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Constant temperature model for nuclear level density
Physics Letters B 783 (2018) 428–433
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Physics Letters B www.elsevier.com/locate/physletb
Constant temperature model for nuclear level density Vladimir Zelevinsky a,b , Sofia Karampagia a,c,∗ , Alexander Berlaga d a
National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824-1321, USA Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824-2320, USA Department of Physics, Grand Valley State University, Allendale, MI 49504, USA d Henry M. Gunn High School, Palo Alto, CA 94306, USA b c
a r t i c l e
i n f o
Article history: Received 22 March 2018 Received in revised form 9 July 2018 Accepted 11 July 2018 Available online 17 July 2018 Editor: J.-P. Blaizot
a b s t r a c t We study physics related to the nuclear level density calculated either in a realistic shell model or, equivalently, with the use of the statistical moments method. At excitation energy up to 12–15 MeV, the obtained level density grows exponentially being well described by the so-called constant temperature model. We discuss the physical meaning of the effective temperature parameter and its dependence on the interaction Hamiltonian including nucleon pairing and deformation effects. The possible interpretation relates the underlying physics with the gradual chaotization of typical wave functions rather than with the pairing phase transition. © 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .
1. Introduction The importance of reliable knowledge of the nuclear level density is obvious for the understanding of nuclear reactions − in the laboratory, in technological applications, and in cosmos. There is a long history of various approaches to the problem of level density which cannot be fully reflected in this Introduction. We just briefly mention the Fermi-gas description [1–3], more advanced mean-field methods [4] accounting for the pairing correlations, and Monte-Carlo approaches [5]. Of course, modern shell-model (configuration interaction) theory in principle gives, for the accepted Hamiltonian, the exact result [6,7] limited in energy by the unavoidable truncation of the orbital space but this approach is always related to the diagonalization of prohibitively large matrices. The shell-model Monte-Carlo method [8] gives the results without diagonalization but it currently accounts only for the most regular parts of accepted interactions. It was earlier shown [9] that the interaction matrix elements corresponding to incoherent collision-like processes are equally contributing to the resulting level density increasing its width and providing its smooth energy dependence. Modern versions of the shell model using the spectroscopically tested Hamiltonians allow us to predict reliably the level density up to excitation energy of about 12–15 MeV where the resonances
*
Corresponding author at: Department of Physics, Grand Valley State University, Allendale, MI 49504 USA. E-mail address: [email protected] (S. Karampagia).
in the continuum are still not too broad. This could be sufficient for many practical purposes. It turns out, see [9–11] and references therein, that frequently it is possible to avoid the full diagonalization using the so-called moments method [12] based on statistical properties of many-body wave functions. It was shown repeatedly that, in those cases which practically allow the complete diagonalization, results of the moments method are essentially identical at energies of interest to the results from the full shell-model solution. In such cases, the comparison with the experimental data becomes in fact a quality check for the underlying shell-model Hamiltonian. In what follows we discuss some features of physics that determine the level density in a nucleus as an isolated quantum system of strongly interacting fermions. It turns out that in the majority of cases in the energy region of interest, that includes the beginning of the continuum where the energy levels become (still non-overlapping) resonances, the level density grows exponentially. A large systematics of experimental data for many nuclei and comparison with the Fermi-gas approaches can be found in Refs. [9,13,14]. The studies in the shell-model framework [9,11,15] seem to prefer the description usually associated with the socalled constant temperature model where the effective temperature T is introduced as an inverse coefficient in the exponent of the level density as a function of excitation energy. Varying the shellmodel Hamiltonian we can determine the dependence of this parameter on various interaction parts. This puts limitations on the possible interpretation of the constant temperature model. In fact, the parameter T in the level density is rather an analog of the limiting Hagedorn temperature in particle physics [16–18]. The for-
https://doi.org/10.1016/j.physletb.2018.07.023 0370-2693/© 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .
V. Zelevinsky et al. / Physics Letters B 783 (2018) 428–433
mally found thermodynamic temperature T t−d starts at zero for the ground state and approaches the value of T at excitation energy E noticeably higher than T , usually around or above 10 MeV. As in particle physics, this does not mean that the system cannot be heated further but it can be interpreted as a transition to the chaotic stage of more or less randomly interacting constituents. As known from many studies, the internal states of nuclei at such energy are close to random superpositions of many quasiparticle excitations so that the local structure of the spectra can be well juxtaposed to that of the Gaussian orthogonal ensemble [19,20]. At the same time, the obvious interpretation of this evolution as a phase transition from superfluid paired dynamics to the normal Fermi liquid is not sufficient. The specific examples below show that the behavior persists if the standard attractive pairing interaction (the source of nuclear superfluidity) is removed from the Hamiltonian or even substituted by repulsion. Supposedly we have to deal with a more general process of stochastization of dynamics as a typical feature of quantum many-body systems. 2. Constant temperature model For all sd-nuclei, and all classes of states with different values of nuclear spin J , the level density was calculated and tabulated [11] using the shell model USDB Hamiltonian [21] and either the moments method or the full diagonalization; examples for heavier nuclei are given also in [9]. In the majority of cases, the resulting level density can be well described by the so-called constant temperature formula
ρ ( E ) = ρ0 e E /T ,
(1)
where the prefactor is usually written in the form
ρ0 =
1 − E 0 /T . e T
(2)
Here we introduce two parameters, T and E 0 , while E in eq. (1) is the actual nuclear excitation energy counted from the ground state. This parametrization was suggested long ago [22,17] and successfully used for the description of data [9,13–15,18]. In traditional √ Fermi-gas models, the level density typically grows as exp( 2aE ) with the constant a determined by the single-particle level density at the Fermi surface. Fig. 1 shows the evolution of the effective temperature T along the isotope chains of magnesium, aluminum, and silicon [11], while Fig. 2 illustrates the quality of description in the moments method and the USD B shell model Hamiltonian for 24 Mg when compared to the experimental level densities. As discussed earlier [9,11], the parameter T of eq. (1) reaches its minimum at N = Z or at neighboring odd nuclei. This quantity is kind of effective temperature kept constant within a broad interval of excitation energies E. This is the source of the name “constant temperature model”, although the definition (1) just provides 1/ T as the constant rate of increase of the level density as a function of excitation energy. As shown in [11], such a phenomenological expression is indeed working universally for almost all sd-nuclei described by the shell model. It provides the good description in the p f -region as well. Here we have to stress that the effective temperature parameter T in eqs. (1) and (2) does not coincide with the temperature T t−d found from thermodynamics for the system with the level density (1). Indeed, defining microcanonical thermodynamic entropy S through the cumulative level number
E N (E) = 0
dE ρ ( E ) = e S ,
(3)
we come to
S = ln
429
ρ0 T (e E /T − 1) .
(4)
As always for a system with a discrete energy spectrum, this expression violating the third law of thermodynamics acquires the meaning only at non-zero (practically quite small) excitation energy, when it makes sense to speak about the level density. Now we can introduce the thermodynamic temperature,
T t−d =
∂S ∂E
−1
= T 1 − e− E /T ,
(5)
which is always lower than our auxiliary temperature T but coincides with that at E T . The thermodynamic temperature T t−d starts from zero at very low excitation energy and then grows as a function of E to the maximum value of T , while the effective temperature T is constant in the broad interval of excitation energies, usually including the continuum threshold. The thermodynamic heat capacity ∂ E /∂ T t−d = exp( E / T ) increases from E = 0 exponentially (in usual Fermi-gas models it grows linearly). As known from discussions of the Hagedorn temperature extracted from the exponentially growing density of resonances, eq. (5) does not mean the existence of the absolute hottest temperature. The system just becomes a chaotic gas of randomly interacting constituents (quarks or strings in quantum field theory and quasiparticles in the nuclear case). At higher excitation energy the exponential level density law (1) does not work anymore; the partition function defined in a standard way,
Tr(e − H / T t−d ) =
dE ρ ( E )e − E / T t−d ,
(6)
would diverge. It was shown long ago, see for example [19], that the full shell-model level density in a finite fermionic space is given by a particle–hole symmetric bell-shape curve that is essentially Gaussian close to the centroid; the going down part beyond the energy centroid formally corresponds to negative temperature (inversion of occupancies in the finite Hilbert space). Above some energy, realistically much lower than the Gaussian centroid, the validity of the description (1) expires, even if the states outside of the originally truncated orbital space still do not enter the game. From Fig. 3, one can see the excitation energy limits (∼ 15 MeV) for the constant temperature model applied to 24 Mg. Assuming for the global shell-model level density the standard Gaussian shape,
ρg (E ) = √
1 2πσ 2
2 2 e −( E − E c ) /(2σ ) ,
(7)
we should be able to match continuously this function with the constant temperature model valid at low excitation energy. The global description of the Gaussian (7) introduces [19] an average temperature
T g (E) =
σ2 Ec − E
.
(8)
This shows the infinite temperature at the centroid with the jump to negative temperature after the middle which physically displays the particle–hole symmetry in a finite orbital space. As shown long ago [19,23], the formally defined global temperature T g with its Gaussian energy dependence agrees with the temperature parameter fit by the fermionic occupation numbers of individual states found in the exact shell-model solution (see also the discussion in [24] combining atomic and nuclear examples). This means that, starting with some excitation energy, the
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Fig. 3. The shell-model level density for 24 Mg (dotted line) and the constant temperature (CT) model of eq. (1), solid line, with parameters T = 3.43 MeV and E 0 = 1.65 MeV for a fitting range 2–15 MeV.
3. Effective temperature and pairing interaction Fig. 1. Effective temperature parameter T for the isotopes of magnesium, aluminum, and silicon. The level density is calculated with the USDB version of the shell model.
Fig. 2. Comparison of experimental nuclear level density for positive parity states (green stair line) with the analogous level density calculated with the USDB shell model (blue dashed stair line) and the moments method (solid red line) for 24 Mg. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
system is analogous to the heated Fermi-liquid with chaotically complicated many-body wave functions and average occupancies given by the Fermi-statistics. If so, at some excitation energy, the effective temperature T well describing the initial stage of the evolution, should agree with the global temperature T g . This indeed takes place. For example, for the states J = 0+ in 28 Si, the parameter T from the constant temperature fit equals 3.11 MeV [11]. At a typical energy inside this region, for example E = 10 MeV, we can use the global shell-model results [19], σ ≈ 11 MeV and E c ≈ 50 MeV, to get T g = 3.03 MeV and, according to Eq. (5), T t−d ≈ 0.96 T ≈ 3.0 MeV. Although the full comparison for all cases should be done in the future, we see that the constant temperature model reasonably corresponds to the limiting situation when the thermalization is not finished yet and the system performs a kind of the crossover transition to a chaotic stage.
It follows from several studies [9–11,25] that the effective temperature T is theoretically determined by the parameters of the shell model, single-particle energies and residual interactions, see the results for few nuclides in Fig. 1. As mentioned above, the minimum of this parameter within a given isotope chain corresponds to N = Z for even–even sd-nuclei (like 28 Si and 24 Mg) or to the closest isotopes with N = Z ± 1 for aluminum isotopes. With thermodynamic temperature always lower than T , these systems are stronger “frozen” at low energy but faster increase their level densities with excitation energy. This immediately suggests an important role of pairing correlations that was put in the foundation of the entire constant-temperature concept for nuclei by Moretto [17,18]. In addition, quartic and other correlations may play a role in the evolution of the parameter T near the N = Z line when the dynamics includes the lowest isospin values. The shell-model formalism allows us to explicitly check the underlying physics. The shell-model version [21] used here contains, for the sd-space, three single-particle energies and 63 matrix elements of two-body interaction. Six of those matrix elements correspond to the isospin-invariant (isospin 1) pairing interactions. The total Hamiltonian can be presented [9] as
H = h + k1 V 1 + k2 V 2 ,
(9)
where h provides single-particle energies, V 1 contains only six pairing matrix elements while V 2 is composed of all remaining two-body matrix elements. We can vary coefficients k1 and k2 and follow the change of the resulting level density that requires the reassignment of the temperature parameter T . As shown earlier [9], the part V 2 is necessary for correct physical description: due to these interactions, which include incoherent collision-like processes, the level density as a function of energy acquires a smooth shape without rough bumps reflecting the subshell structures. The realistic case corresponds to k1 = k2 = 1. Fig. 4 shows the change of the parameter T as a function of the pairing strength k1 (at k2 = 1 fixed) for the magnesium and aluminum isotopes, while Fig. 5 shows the change of the parameter T as a function of k2 , at the fixed proper value k1 = 1. In all cases the calculated level density can be well described by the constant temperature model using the values of the parameter T fit to individual isotopes. Even with repulsive pairing, k1 < 0, see Fig. 6, the constant temperature model fits well the calculated level density.
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Fig. 4. Evolution of the effective temperature parameter T under variation of the pairing strength k1 for the isotopes of magnesium and aluminum.
Fig. 6. Moments method level density for 24 Mg (red slanted line) and the level density given by the constant temperature formula when the latter is fitted on the moments method level density (black line) for (a) the lowest pairing strength (k1 = 0.1) and (b) the highest pairing strength (k1 = 1.3).
behavior in other matrix elements. In general, larger values of k2 make statistical regularities of the level density more pronounced. As seen in Fig. 6 the fit of the level densities provided by the moments method with the constant temperature formula is excellent both for low and high pairing strength. This situation is not restricted in 24 Mg, the picture is the same for all nuclei studied. As an additional measure of the quality of the fit, we calculated the error factor f err [26–28], defined in Eq. (10), which is a measure of the deviation of the calculated level density using the moments method (ρmm ) from the constant temperature level density (ρct ),
f err Fig. 5. Evolution of the effective temperature parameter T under variation of the non-pairing interactions strength k2 for the isotopes of magnesium and aluminum.
In this region, the parameter T does not depend on negative k1 being in some cases slightly higher than for the attractive pairing (the system is less frozen). As seen from Fig. 4, the effective temperature slowly increases as k1 decreases. The stronger attractive pairing (greater k1 ) leads to stronger freezing but the total fall of T due to the increase of attractive pairing is quite small. The similar behavior for practically all cases above A = 21 demonstrates the generality of this effect. Therefore we have to exclude the standard isospin-1 pairing as a sole culprit behind the constant temperature model even if it still plays a noticeable role. As a function of k2 , the effective temperature, Fig. 5, grows when the interaction coming from the non-pairing matrix elements increases. We notice in Fig. 4 that the complete absence of pairing interaction does not affect the appearance of the minimum of the temperature at N = Z and N = Z ± 1 for magnesium and aluminum, respectively. On the other hand, the strong presence of the matrix elements not related to pairing (greater k2 ) is necessary to form the minimum of the temperature as a function of N − Z , as seen in Fig. 5. Consequently, the pairing interaction is not solely responsible for the minimum in temperature at N = Z and N = Z ± 1 and one should look for additional sources of this
⎞ ⎛ Ni 1 ρ mm ( E i ) 2
⎠, = exp ⎝ ln Ni ρct ( E i )
(10)
i =1
where N i is the number of the discrete energy intervals and E i the excitation energy. We found that for 24 Mg the deviation for the lowest pairing strength starts at f err = 1.54 and slowly decreases, reaching f err = 1.49 for the realistic interaction and falling to f err = 1.42 at the highest pairing strength (k1 = 1.3). Obviously, this error is really small proving that the fit is excellent for all cases. Again, this conclusion applies to all nuclei. Additionally, we fitted the moments method level density of 24 Mg, for the realistic interaction, with the Bethe formula [3]
ρ (U ) =
exp[2
1
√
12 2σ
2
√
αU ] , α 1/4 U 5/4
(11)
√ 2/ 3
where, U = E − δ and σ 2 = 0.0888 A α ( E − δ). We found that the f err for the realistic interaction is 1.38, for the fitting range of 2–15 MeV, i.e. equally good as the constant temperature formula fitting. Both the constant temperature and Bethe formula describe well the level density when the fitting takes place at small energy intervals and any deviations between the two are not noticeable. A detailed comparison of the constant temperature model and the Bethe formula should be a topic of future studies. It is important to understand what happens in separate angular momentum classes of eigenstates. The typical behavior is
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Fig. 7. Evolution of the effective temperature parameter T under variation of k1 and k2 for the states J = 0 and J = 2 of 28,30 Si.
similar for different nuclear spins J although, probably, a more detailed analysis would be able to establish some specific features for individual values of J but on a smaller scale than the generic behavior. Examples for J = 0 and J = 2 are shown in Fig. 7. The effective temperature is constant even at k1 < 0 (antipairing). As a function of growing positive k1 , the effective temperature is still almost constant for J = 0 but increases a little for 28,30 Si when we take pairing anomalously strong, k1 > 1. This probably reflects increased population of a higher orbital, in this case d3/2 , by stronger pairing. For J = 2, the effective temperature is still constant at k1 < 0 but at a lower value than for J = 0 (mainly rotational band-head states in the unpaired system, with a higher moment of inertia and therefore lower rotational energy). For very strong pairing, k1 > +1, the effective temperature goes down, typically lower than for J = 0. This is accompanied by the growth of the increase rate of the level density in the situation of its collective enhancement discussed in this approach in Ref. [10]. In general, the J -dependence deserves to be studied in more detail for various nuclei along with the appropriate analysis of actual low-lying states. 4. Effective temperature and nuclear shape The concept of the collective enhancement of the level density is well known [29]. In a nucleus with clearly developed lowenergy collective degrees of freedom, especially rotations, every internal configuration becomes a band head of a collective sequence of states increasing therefore the level density at low energy. Of course, this just redistributes the levels along the spectrum without changing their total number. Varying the Hamiltonian matrix elements mainly responsible for the onset of deformation [25] we can artificially induce a shape transition and therefore effectively move the whole ladder of energy levels down revealing the collective enhancement. It was earlier shown in the framework of the shell model [30] that the transition to deformation is mainly regulated by the part of the two-body residual interaction that mixes the single-particle orbitals with the selection rule for the orbital momentum | | = 2 and the same parity. Then the nucleus undergoes the quadrupole deformation of the mean field and corresponding evolution of observables. This was confirmed by the analysis of the changes of the level density induced by the same variation of the Hamiltonian in
Fig. 8. Evolution of the effective temperature parameter T under variation of λ for the isotopes of magnesium and aluminum.
even–even [25] and odd- A and odd–odd [10] nuclei. Now we can study what is going on with the effective temperature parameter under such a transformation. Here it is convenient to separate from the shell-model twobody interactions the set of matrix elements U 1 that change a spherical orbital of one particle (in the sd-space mixing s- and d-states of one of colliding particles). With the rest of matrix elements combined in U 2 , we have
H = h + λU 1 + U 2 .
(12)
As in the pairing case, the constant temperature model still works fine; the resulting behavior of the effective temperature is shown in Fig. 8 for the isotopes of magnesium and aluminum. There is a clear appearance of a minimum in the parameter T at N = Z and N = Z ± 1 isotopes, irrespectively of how small or large the contribution of the U 1 matrix elements is. As a function of λ, a clear general feature is the systematic minimum of T close to λ = 0. The growth of T with increasing |λ| is usually symmetric with respect to the sign of λ that shows the thermal equivalence of trends to oblate or prolate deformation. In both cases, the appearance of static deformation and/or low-energy vibrational modes results in the collective enhancement of the level density discussed in [10]. The higher density of collective states and rotational/vibrational bands built on them corresponds to higher effective temperature and therefore the slower relative rate, 1/ T = (1/ρ )(dρ /dE ) of the level density growing in the direction to mixing (band crossing) and chaotization. The line of λ-dependence in even–even isotopes of magnesium and silicon keeps the lowest level for the case N = Z stressing again the additional low-energy features available for such nuclei. 5. Conclusion The constant temperature model for the nuclear level density turns out to work well at not very high excitation energy reproducing the results of the exact solution of the nuclear many-body problem in the framework of the configuration interaction (sd shell model in this specific case). For a large configuration space, the statistical moments method based on chaotic properties of the interacting many-body Fermi-system allows to derive the level density avoiding full diagonalization. The practical applicability of this method, and therefore of the obtained level density, is limited by
V. Zelevinsky et al. / Physics Letters B 783 (2018) 428–433
the truncations used on the stage of the derivation and fitting the shell-model parameters. Here one important general deficiency of the configuration interaction approach (not only the traditional shell model) can come to the surface: the level density feels many small matrix elements which could be not significant for the spectroscopic results, such as the energy spectrum and main transition rates, but needed to smooth the resulting level density. One perspective direction of studies would be to substitute those small matrix elements (not defined by the simple low-energy observables) by the random forces of average strength. Such forces are expected to change smoothly from one nucleus to another. This should be studied separately. In the exact shell-model solution for a finite configuration space, the gross behavior of the level density is characterized by the Gaussian. At relatively low excitation energy, where we expect detailed validity of the shell model within its truncated space, the level density, as it was argued above, is well described by the exponential function of energy with a constant temperature parameter T . This effective temperature does not coincide with the usual thermodynamic temperature T t−d . The latter can be defined in such a way that it starts with zero at the ground state and grows to catch with the parameter T . The situation reminds the Hagedorn limiting temperature in particle physics describing the exponential proliferation of resonance states. In a non-relativistic system, as the nucleus, we suggest to describe the situation in terms of the crossover from the ordered low-lying states to quantum chaos locally close to the Gaussian orthogonal ensemble [19]. Then, at some point, the effective temperature matches that of the thermalized chaotic Fermi-liquid. The results of this study show that the situation cannot be described by the simple picture of the pairing phase transition. Although pairing plays a role in the whole picture, the latter is qualitatively the same with no pairing interaction and even with repulsive pairing when the effective temperature just becomes slightly higher. The effective temperature also increases with artificial deformation brought by the corresponding variation of the responsible interaction parameters. With the collective enhancement of the low-lying level density, this means a slower relative growth of the level density along its path to full chaotization. The whole physics is still only surface-like probed and has to be studied in more detail.
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Acknowledgements The authors thank the Facility for Rare Isotope Beams at Michigan State University for support and B.A. Brown for useful discussions. The work was supported by the US NSF grant PHY-1404442. V.Z. acknowledges a useful discussion at the seminar of the Tel Aviv University. A. Berlaga is grateful to the High School Honors Student Program at the Michigan State University (summer of 2017) that provided an opportunity to participate in this study. The collaboration with R.A. Sen’kov at the beginning of this work is gratefully acknowledged. References [1] H. Bethe, Rev. Mod. Phys. 9 (1937) 69. [2] N. Rosenzweig, Phys. Rev. 105 (1957) 950; N. Rosenzweig, Phys. Rev. 108 (1957) 817. [3] A. Gilbert, A.G.W. Cameron, Can. J. Phys. 43 (1965) 1446. [4] S. Goriely, S. Hilaire, A.J. Koning, M. Sin, R. Capote, Phys. Rev. C 79 (2009) 024612. [5] Y. Alhassid, L. Fang, H. Nakada, Phys. Rev. Lett. 101 (2008) 082501. [6] M. Horoi, J. Kaiser, V. Zelevinsky, Phys. Rev. C 67 (2003) 054309. [7] M. Horoi, M. Ghita, V. Zelevinsky, Phys. Rev. C 69 (2004), 041307(R). [8] Y. Alhassid, S. Liu, H. Nakada, Phys. Rev. Lett. 99 (2007) 162504. [9] R.A. Sen’kov, V. Zelevinsky, Phys. Rev. C 93 (2016) 064304. [10] S. Karampagia, A. Renzaglia, V. Zelevinsky, Nucl. Phys. A 962 (2017) 46. [11] S. Karampagia, R.A. Sen’kov, V. Zelevinsky, At. Data Nucl. Data Tables 120 (2018) 1. [12] R.A. Sen’kov, M. Horoi, V. Zelevinsky, Comput. Phys. Commun. 184 (2013) 215. [13] T. von Egidy, D. Bucurescu, Phys. Rev. C 72 (2005) 044311. [14] D. Bucurescu, T. von Egidy, Phys. Rev. C 72 (2005) 067304. [15] M. Horoi, J. Dissanayake, arXiv:1706.05391. [16] J.J. Atick, E. Witten, Nucl. Phys. B 3210 (1988) 291. [17] L.G. Moretto, Nucl. Phys. A 243 (1975) 77. [18] L.G. Moretto, A.C. Larsen, M. Guttormsen, S. Siem, AIP Conf. Proc. 1681 (2015) 040011. [19] V. Zelevinsky, B.A. Brown, N. Frazier, M. Horoi, Phys. Rep. 276 (1996) 315. [20] H.A. Weidenmüller, G.E. Mitchell, Rev. Mod. Phys. 81 (2009) 539. [21] B.A. Brown, B.H. Wildenthal, Annu. Rev. Nucl. Part. Sci. 38 (1988) 29. [22] T. Ericson, Adv. Phys. 9 (1960) 425. [23] V. Zelevinsky, Annu. Rev. Nucl. Part. Sci. 46 (1996) 237. [24] F. Borgonovi, F.M. Izrailev, L.F. Santos, V.G. Zelevinsky, Phys. Rep. 626 (2016) 1. [25] S. Karampagia, V. Zelevinsky, Phys. Rev. C 94 (2016) 014321. [26] P. Demetriou, S. Goriely, Nucl. Phys. A 695 (2001) 95. [27] S. Goriely, S. Hilaire, A.J. Koning, Phys. Rev. C 78 (2008) 064307. [28] M. Scott, M. Horoi, Europhys. Lett. 91 (2010) 52001. [29] A.V. Ignatyuk, K.K. Istekov, G.N. Smirenkin, Sov. J. Nucl. Phys. 29 (1979) 450. [30] M. Horoi, V. Zelevinsky, Phys. Rev. C 81 (2010) 034306.
Contents lists available at ScienceDirect
Physics Letters B www.elsevier.com/locate/physletb
Constant temperature model for nuclear level density Vladimir Zelevinsky a,b , Sofia Karampagia a,c,∗ , Alexander Berlaga d a
National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824-1321, USA Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824-2320, USA Department of Physics, Grand Valley State University, Allendale, MI 49504, USA d Henry M. Gunn High School, Palo Alto, CA 94306, USA b c
a r t i c l e
i n f o
Article history: Received 22 March 2018 Received in revised form 9 July 2018 Accepted 11 July 2018 Available online 17 July 2018 Editor: J.-P. Blaizot
a b s t r a c t We study physics related to the nuclear level density calculated either in a realistic shell model or, equivalently, with the use of the statistical moments method. At excitation energy up to 12–15 MeV, the obtained level density grows exponentially being well described by the so-called constant temperature model. We discuss the physical meaning of the effective temperature parameter and its dependence on the interaction Hamiltonian including nucleon pairing and deformation effects. The possible interpretation relates the underlying physics with the gradual chaotization of typical wave functions rather than with the pairing phase transition. © 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .
1. Introduction The importance of reliable knowledge of the nuclear level density is obvious for the understanding of nuclear reactions − in the laboratory, in technological applications, and in cosmos. There is a long history of various approaches to the problem of level density which cannot be fully reflected in this Introduction. We just briefly mention the Fermi-gas description [1–3], more advanced mean-field methods [4] accounting for the pairing correlations, and Monte-Carlo approaches [5]. Of course, modern shell-model (configuration interaction) theory in principle gives, for the accepted Hamiltonian, the exact result [6,7] limited in energy by the unavoidable truncation of the orbital space but this approach is always related to the diagonalization of prohibitively large matrices. The shell-model Monte-Carlo method [8] gives the results without diagonalization but it currently accounts only for the most regular parts of accepted interactions. It was earlier shown [9] that the interaction matrix elements corresponding to incoherent collision-like processes are equally contributing to the resulting level density increasing its width and providing its smooth energy dependence. Modern versions of the shell model using the spectroscopically tested Hamiltonians allow us to predict reliably the level density up to excitation energy of about 12–15 MeV where the resonances
*
Corresponding author at: Department of Physics, Grand Valley State University, Allendale, MI 49504 USA. E-mail address: [email protected] (S. Karampagia).
in the continuum are still not too broad. This could be sufficient for many practical purposes. It turns out, see [9–11] and references therein, that frequently it is possible to avoid the full diagonalization using the so-called moments method [12] based on statistical properties of many-body wave functions. It was shown repeatedly that, in those cases which practically allow the complete diagonalization, results of the moments method are essentially identical at energies of interest to the results from the full shell-model solution. In such cases, the comparison with the experimental data becomes in fact a quality check for the underlying shell-model Hamiltonian. In what follows we discuss some features of physics that determine the level density in a nucleus as an isolated quantum system of strongly interacting fermions. It turns out that in the majority of cases in the energy region of interest, that includes the beginning of the continuum where the energy levels become (still non-overlapping) resonances, the level density grows exponentially. A large systematics of experimental data for many nuclei and comparison with the Fermi-gas approaches can be found in Refs. [9,13,14]. The studies in the shell-model framework [9,11,15] seem to prefer the description usually associated with the socalled constant temperature model where the effective temperature T is introduced as an inverse coefficient in the exponent of the level density as a function of excitation energy. Varying the shellmodel Hamiltonian we can determine the dependence of this parameter on various interaction parts. This puts limitations on the possible interpretation of the constant temperature model. In fact, the parameter T in the level density is rather an analog of the limiting Hagedorn temperature in particle physics [16–18]. The for-
https://doi.org/10.1016/j.physletb.2018.07.023 0370-2693/© 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .
V. Zelevinsky et al. / Physics Letters B 783 (2018) 428–433
mally found thermodynamic temperature T t−d starts at zero for the ground state and approaches the value of T at excitation energy E noticeably higher than T , usually around or above 10 MeV. As in particle physics, this does not mean that the system cannot be heated further but it can be interpreted as a transition to the chaotic stage of more or less randomly interacting constituents. As known from many studies, the internal states of nuclei at such energy are close to random superpositions of many quasiparticle excitations so that the local structure of the spectra can be well juxtaposed to that of the Gaussian orthogonal ensemble [19,20]. At the same time, the obvious interpretation of this evolution as a phase transition from superfluid paired dynamics to the normal Fermi liquid is not sufficient. The specific examples below show that the behavior persists if the standard attractive pairing interaction (the source of nuclear superfluidity) is removed from the Hamiltonian or even substituted by repulsion. Supposedly we have to deal with a more general process of stochastization of dynamics as a typical feature of quantum many-body systems. 2. Constant temperature model For all sd-nuclei, and all classes of states with different values of nuclear spin J , the level density was calculated and tabulated [11] using the shell model USDB Hamiltonian [21] and either the moments method or the full diagonalization; examples for heavier nuclei are given also in [9]. In the majority of cases, the resulting level density can be well described by the so-called constant temperature formula
ρ ( E ) = ρ0 e E /T ,
(1)
where the prefactor is usually written in the form
ρ0 =
1 − E 0 /T . e T
(2)
Here we introduce two parameters, T and E 0 , while E in eq. (1) is the actual nuclear excitation energy counted from the ground state. This parametrization was suggested long ago [22,17] and successfully used for the description of data [9,13–15,18]. In traditional √ Fermi-gas models, the level density typically grows as exp( 2aE ) with the constant a determined by the single-particle level density at the Fermi surface. Fig. 1 shows the evolution of the effective temperature T along the isotope chains of magnesium, aluminum, and silicon [11], while Fig. 2 illustrates the quality of description in the moments method and the USD B shell model Hamiltonian for 24 Mg when compared to the experimental level densities. As discussed earlier [9,11], the parameter T of eq. (1) reaches its minimum at N = Z or at neighboring odd nuclei. This quantity is kind of effective temperature kept constant within a broad interval of excitation energies E. This is the source of the name “constant temperature model”, although the definition (1) just provides 1/ T as the constant rate of increase of the level density as a function of excitation energy. As shown in [11], such a phenomenological expression is indeed working universally for almost all sd-nuclei described by the shell model. It provides the good description in the p f -region as well. Here we have to stress that the effective temperature parameter T in eqs. (1) and (2) does not coincide with the temperature T t−d found from thermodynamics for the system with the level density (1). Indeed, defining microcanonical thermodynamic entropy S through the cumulative level number
E N (E) = 0
dE ρ ( E ) = e S ,
(3)
we come to
S = ln
429
ρ0 T (e E /T − 1) .
(4)
As always for a system with a discrete energy spectrum, this expression violating the third law of thermodynamics acquires the meaning only at non-zero (practically quite small) excitation energy, when it makes sense to speak about the level density. Now we can introduce the thermodynamic temperature,
T t−d =
∂S ∂E
−1
= T 1 − e− E /T ,
(5)
which is always lower than our auxiliary temperature T but coincides with that at E T . The thermodynamic temperature T t−d starts from zero at very low excitation energy and then grows as a function of E to the maximum value of T , while the effective temperature T is constant in the broad interval of excitation energies, usually including the continuum threshold. The thermodynamic heat capacity ∂ E /∂ T t−d = exp( E / T ) increases from E = 0 exponentially (in usual Fermi-gas models it grows linearly). As known from discussions of the Hagedorn temperature extracted from the exponentially growing density of resonances, eq. (5) does not mean the existence of the absolute hottest temperature. The system just becomes a chaotic gas of randomly interacting constituents (quarks or strings in quantum field theory and quasiparticles in the nuclear case). At higher excitation energy the exponential level density law (1) does not work anymore; the partition function defined in a standard way,
Tr(e − H / T t−d ) =
dE ρ ( E )e − E / T t−d ,
(6)
would diverge. It was shown long ago, see for example [19], that the full shell-model level density in a finite fermionic space is given by a particle–hole symmetric bell-shape curve that is essentially Gaussian close to the centroid; the going down part beyond the energy centroid formally corresponds to negative temperature (inversion of occupancies in the finite Hilbert space). Above some energy, realistically much lower than the Gaussian centroid, the validity of the description (1) expires, even if the states outside of the originally truncated orbital space still do not enter the game. From Fig. 3, one can see the excitation energy limits (∼ 15 MeV) for the constant temperature model applied to 24 Mg. Assuming for the global shell-model level density the standard Gaussian shape,
ρg (E ) = √
1 2πσ 2
2 2 e −( E − E c ) /(2σ ) ,
(7)
we should be able to match continuously this function with the constant temperature model valid at low excitation energy. The global description of the Gaussian (7) introduces [19] an average temperature
T g (E) =
σ2 Ec − E
.
(8)
This shows the infinite temperature at the centroid with the jump to negative temperature after the middle which physically displays the particle–hole symmetry in a finite orbital space. As shown long ago [19,23], the formally defined global temperature T g with its Gaussian energy dependence agrees with the temperature parameter fit by the fermionic occupation numbers of individual states found in the exact shell-model solution (see also the discussion in [24] combining atomic and nuclear examples). This means that, starting with some excitation energy, the
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Fig. 3. The shell-model level density for 24 Mg (dotted line) and the constant temperature (CT) model of eq. (1), solid line, with parameters T = 3.43 MeV and E 0 = 1.65 MeV for a fitting range 2–15 MeV.
3. Effective temperature and pairing interaction Fig. 1. Effective temperature parameter T for the isotopes of magnesium, aluminum, and silicon. The level density is calculated with the USDB version of the shell model.
Fig. 2. Comparison of experimental nuclear level density for positive parity states (green stair line) with the analogous level density calculated with the USDB shell model (blue dashed stair line) and the moments method (solid red line) for 24 Mg. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
system is analogous to the heated Fermi-liquid with chaotically complicated many-body wave functions and average occupancies given by the Fermi-statistics. If so, at some excitation energy, the effective temperature T well describing the initial stage of the evolution, should agree with the global temperature T g . This indeed takes place. For example, for the states J = 0+ in 28 Si, the parameter T from the constant temperature fit equals 3.11 MeV [11]. At a typical energy inside this region, for example E = 10 MeV, we can use the global shell-model results [19], σ ≈ 11 MeV and E c ≈ 50 MeV, to get T g = 3.03 MeV and, according to Eq. (5), T t−d ≈ 0.96 T ≈ 3.0 MeV. Although the full comparison for all cases should be done in the future, we see that the constant temperature model reasonably corresponds to the limiting situation when the thermalization is not finished yet and the system performs a kind of the crossover transition to a chaotic stage.
It follows from several studies [9–11,25] that the effective temperature T is theoretically determined by the parameters of the shell model, single-particle energies and residual interactions, see the results for few nuclides in Fig. 1. As mentioned above, the minimum of this parameter within a given isotope chain corresponds to N = Z for even–even sd-nuclei (like 28 Si and 24 Mg) or to the closest isotopes with N = Z ± 1 for aluminum isotopes. With thermodynamic temperature always lower than T , these systems are stronger “frozen” at low energy but faster increase their level densities with excitation energy. This immediately suggests an important role of pairing correlations that was put in the foundation of the entire constant-temperature concept for nuclei by Moretto [17,18]. In addition, quartic and other correlations may play a role in the evolution of the parameter T near the N = Z line when the dynamics includes the lowest isospin values. The shell-model formalism allows us to explicitly check the underlying physics. The shell-model version [21] used here contains, for the sd-space, three single-particle energies and 63 matrix elements of two-body interaction. Six of those matrix elements correspond to the isospin-invariant (isospin 1) pairing interactions. The total Hamiltonian can be presented [9] as
H = h + k1 V 1 + k2 V 2 ,
(9)
where h provides single-particle energies, V 1 contains only six pairing matrix elements while V 2 is composed of all remaining two-body matrix elements. We can vary coefficients k1 and k2 and follow the change of the resulting level density that requires the reassignment of the temperature parameter T . As shown earlier [9], the part V 2 is necessary for correct physical description: due to these interactions, which include incoherent collision-like processes, the level density as a function of energy acquires a smooth shape without rough bumps reflecting the subshell structures. The realistic case corresponds to k1 = k2 = 1. Fig. 4 shows the change of the parameter T as a function of the pairing strength k1 (at k2 = 1 fixed) for the magnesium and aluminum isotopes, while Fig. 5 shows the change of the parameter T as a function of k2 , at the fixed proper value k1 = 1. In all cases the calculated level density can be well described by the constant temperature model using the values of the parameter T fit to individual isotopes. Even with repulsive pairing, k1 < 0, see Fig. 6, the constant temperature model fits well the calculated level density.
V. Zelevinsky et al. / Physics Letters B 783 (2018) 428–433
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Fig. 4. Evolution of the effective temperature parameter T under variation of the pairing strength k1 for the isotopes of magnesium and aluminum.
Fig. 6. Moments method level density for 24 Mg (red slanted line) and the level density given by the constant temperature formula when the latter is fitted on the moments method level density (black line) for (a) the lowest pairing strength (k1 = 0.1) and (b) the highest pairing strength (k1 = 1.3).
behavior in other matrix elements. In general, larger values of k2 make statistical regularities of the level density more pronounced. As seen in Fig. 6 the fit of the level densities provided by the moments method with the constant temperature formula is excellent both for low and high pairing strength. This situation is not restricted in 24 Mg, the picture is the same for all nuclei studied. As an additional measure of the quality of the fit, we calculated the error factor f err [26–28], defined in Eq. (10), which is a measure of the deviation of the calculated level density using the moments method (ρmm ) from the constant temperature level density (ρct ),
f err Fig. 5. Evolution of the effective temperature parameter T under variation of the non-pairing interactions strength k2 for the isotopes of magnesium and aluminum.
In this region, the parameter T does not depend on negative k1 being in some cases slightly higher than for the attractive pairing (the system is less frozen). As seen from Fig. 4, the effective temperature slowly increases as k1 decreases. The stronger attractive pairing (greater k1 ) leads to stronger freezing but the total fall of T due to the increase of attractive pairing is quite small. The similar behavior for practically all cases above A = 21 demonstrates the generality of this effect. Therefore we have to exclude the standard isospin-1 pairing as a sole culprit behind the constant temperature model even if it still plays a noticeable role. As a function of k2 , the effective temperature, Fig. 5, grows when the interaction coming from the non-pairing matrix elements increases. We notice in Fig. 4 that the complete absence of pairing interaction does not affect the appearance of the minimum of the temperature at N = Z and N = Z ± 1 for magnesium and aluminum, respectively. On the other hand, the strong presence of the matrix elements not related to pairing (greater k2 ) is necessary to form the minimum of the temperature as a function of N − Z , as seen in Fig. 5. Consequently, the pairing interaction is not solely responsible for the minimum in temperature at N = Z and N = Z ± 1 and one should look for additional sources of this
⎞ ⎛ Ni 1 ρ mm ( E i ) 2
⎠, = exp ⎝ ln Ni ρct ( E i )
(10)
i =1
where N i is the number of the discrete energy intervals and E i the excitation energy. We found that for 24 Mg the deviation for the lowest pairing strength starts at f err = 1.54 and slowly decreases, reaching f err = 1.49 for the realistic interaction and falling to f err = 1.42 at the highest pairing strength (k1 = 1.3). Obviously, this error is really small proving that the fit is excellent for all cases. Again, this conclusion applies to all nuclei. Additionally, we fitted the moments method level density of 24 Mg, for the realistic interaction, with the Bethe formula [3]
ρ (U ) =
exp[2
1
√
12 2σ
2
√
αU ] , α 1/4 U 5/4
(11)
√ 2/ 3
where, U = E − δ and σ 2 = 0.0888 A α ( E − δ). We found that the f err for the realistic interaction is 1.38, for the fitting range of 2–15 MeV, i.e. equally good as the constant temperature formula fitting. Both the constant temperature and Bethe formula describe well the level density when the fitting takes place at small energy intervals and any deviations between the two are not noticeable. A detailed comparison of the constant temperature model and the Bethe formula should be a topic of future studies. It is important to understand what happens in separate angular momentum classes of eigenstates. The typical behavior is
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Fig. 7. Evolution of the effective temperature parameter T under variation of k1 and k2 for the states J = 0 and J = 2 of 28,30 Si.
similar for different nuclear spins J although, probably, a more detailed analysis would be able to establish some specific features for individual values of J but on a smaller scale than the generic behavior. Examples for J = 0 and J = 2 are shown in Fig. 7. The effective temperature is constant even at k1 < 0 (antipairing). As a function of growing positive k1 , the effective temperature is still almost constant for J = 0 but increases a little for 28,30 Si when we take pairing anomalously strong, k1 > 1. This probably reflects increased population of a higher orbital, in this case d3/2 , by stronger pairing. For J = 2, the effective temperature is still constant at k1 < 0 but at a lower value than for J = 0 (mainly rotational band-head states in the unpaired system, with a higher moment of inertia and therefore lower rotational energy). For very strong pairing, k1 > +1, the effective temperature goes down, typically lower than for J = 0. This is accompanied by the growth of the increase rate of the level density in the situation of its collective enhancement discussed in this approach in Ref. [10]. In general, the J -dependence deserves to be studied in more detail for various nuclei along with the appropriate analysis of actual low-lying states. 4. Effective temperature and nuclear shape The concept of the collective enhancement of the level density is well known [29]. In a nucleus with clearly developed lowenergy collective degrees of freedom, especially rotations, every internal configuration becomes a band head of a collective sequence of states increasing therefore the level density at low energy. Of course, this just redistributes the levels along the spectrum without changing their total number. Varying the Hamiltonian matrix elements mainly responsible for the onset of deformation [25] we can artificially induce a shape transition and therefore effectively move the whole ladder of energy levels down revealing the collective enhancement. It was earlier shown in the framework of the shell model [30] that the transition to deformation is mainly regulated by the part of the two-body residual interaction that mixes the single-particle orbitals with the selection rule for the orbital momentum | | = 2 and the same parity. Then the nucleus undergoes the quadrupole deformation of the mean field and corresponding evolution of observables. This was confirmed by the analysis of the changes of the level density induced by the same variation of the Hamiltonian in
Fig. 8. Evolution of the effective temperature parameter T under variation of λ for the isotopes of magnesium and aluminum.
even–even [25] and odd- A and odd–odd [10] nuclei. Now we can study what is going on with the effective temperature parameter under such a transformation. Here it is convenient to separate from the shell-model twobody interactions the set of matrix elements U 1 that change a spherical orbital of one particle (in the sd-space mixing s- and d-states of one of colliding particles). With the rest of matrix elements combined in U 2 , we have
H = h + λU 1 + U 2 .
(12)
As in the pairing case, the constant temperature model still works fine; the resulting behavior of the effective temperature is shown in Fig. 8 for the isotopes of magnesium and aluminum. There is a clear appearance of a minimum in the parameter T at N = Z and N = Z ± 1 isotopes, irrespectively of how small or large the contribution of the U 1 matrix elements is. As a function of λ, a clear general feature is the systematic minimum of T close to λ = 0. The growth of T with increasing |λ| is usually symmetric with respect to the sign of λ that shows the thermal equivalence of trends to oblate or prolate deformation. In both cases, the appearance of static deformation and/or low-energy vibrational modes results in the collective enhancement of the level density discussed in [10]. The higher density of collective states and rotational/vibrational bands built on them corresponds to higher effective temperature and therefore the slower relative rate, 1/ T = (1/ρ )(dρ /dE ) of the level density growing in the direction to mixing (band crossing) and chaotization. The line of λ-dependence in even–even isotopes of magnesium and silicon keeps the lowest level for the case N = Z stressing again the additional low-energy features available for such nuclei. 5. Conclusion The constant temperature model for the nuclear level density turns out to work well at not very high excitation energy reproducing the results of the exact solution of the nuclear many-body problem in the framework of the configuration interaction (sd shell model in this specific case). For a large configuration space, the statistical moments method based on chaotic properties of the interacting many-body Fermi-system allows to derive the level density avoiding full diagonalization. The practical applicability of this method, and therefore of the obtained level density, is limited by
V. Zelevinsky et al. / Physics Letters B 783 (2018) 428–433
the truncations used on the stage of the derivation and fitting the shell-model parameters. Here one important general deficiency of the configuration interaction approach (not only the traditional shell model) can come to the surface: the level density feels many small matrix elements which could be not significant for the spectroscopic results, such as the energy spectrum and main transition rates, but needed to smooth the resulting level density. One perspective direction of studies would be to substitute those small matrix elements (not defined by the simple low-energy observables) by the random forces of average strength. Such forces are expected to change smoothly from one nucleus to another. This should be studied separately. In the exact shell-model solution for a finite configuration space, the gross behavior of the level density is characterized by the Gaussian. At relatively low excitation energy, where we expect detailed validity of the shell model within its truncated space, the level density, as it was argued above, is well described by the exponential function of energy with a constant temperature parameter T . This effective temperature does not coincide with the usual thermodynamic temperature T t−d . The latter can be defined in such a way that it starts with zero at the ground state and grows to catch with the parameter T . The situation reminds the Hagedorn limiting temperature in particle physics describing the exponential proliferation of resonance states. In a non-relativistic system, as the nucleus, we suggest to describe the situation in terms of the crossover from the ordered low-lying states to quantum chaos locally close to the Gaussian orthogonal ensemble [19]. Then, at some point, the effective temperature matches that of the thermalized chaotic Fermi-liquid. The results of this study show that the situation cannot be described by the simple picture of the pairing phase transition. Although pairing plays a role in the whole picture, the latter is qualitatively the same with no pairing interaction and even with repulsive pairing when the effective temperature just becomes slightly higher. The effective temperature also increases with artificial deformation brought by the corresponding variation of the responsible interaction parameters. With the collective enhancement of the low-lying level density, this means a slower relative growth of the level density along its path to full chaotization. The whole physics is still only surface-like probed and has to be studied in more detail.
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Acknowledgements The authors thank the Facility for Rare Isotope Beams at Michigan State University for support and B.A. Brown for useful discussions. The work was supported by the US NSF grant PHY-1404442. V.Z. acknowledges a useful discussion at the seminar of the Tel Aviv University. A. Berlaga is grateful to the High School Honors Student Program at the Michigan State University (summer of 2017) that provided an opportunity to participate in this study. The collaboration with R.A. Sen’kov at the beginning of this work is gratefully acknowledged. References [1] H. Bethe, Rev. Mod. Phys. 9 (1937) 69. [2] N. Rosenzweig, Phys. Rev. 105 (1957) 950; N. Rosenzweig, Phys. Rev. 108 (1957) 817. [3] A. Gilbert, A.G.W. Cameron, Can. J. Phys. 43 (1965) 1446. [4] S. Goriely, S. Hilaire, A.J. Koning, M. Sin, R. Capote, Phys. Rev. C 79 (2009) 024612. [5] Y. Alhassid, L. Fang, H. Nakada, Phys. Rev. Lett. 101 (2008) 082501. [6] M. Horoi, J. Kaiser, V. Zelevinsky, Phys. Rev. C 67 (2003) 054309. [7] M. Horoi, M. Ghita, V. Zelevinsky, Phys. Rev. C 69 (2004), 041307(R). [8] Y. Alhassid, S. Liu, H. Nakada, Phys. Rev. Lett. 99 (2007) 162504. [9] R.A. Sen’kov, V. Zelevinsky, Phys. Rev. C 93 (2016) 064304. [10] S. Karampagia, A. Renzaglia, V. Zelevinsky, Nucl. Phys. A 962 (2017) 46. [11] S. Karampagia, R.A. Sen’kov, V. Zelevinsky, At. Data Nucl. Data Tables 120 (2018) 1. [12] R.A. Sen’kov, M. Horoi, V. Zelevinsky, Comput. Phys. Commun. 184 (2013) 215. [13] T. von Egidy, D. Bucurescu, Phys. Rev. C 72 (2005) 044311. [14] D. Bucurescu, T. von Egidy, Phys. Rev. C 72 (2005) 067304. [15] M. Horoi, J. Dissanayake, arXiv:1706.05391. [16] J.J. Atick, E. Witten, Nucl. Phys. B 3210 (1988) 291. [17] L.G. Moretto, Nucl. Phys. A 243 (1975) 77. [18] L.G. Moretto, A.C. Larsen, M. Guttormsen, S. Siem, AIP Conf. Proc. 1681 (2015) 040011. [19] V. Zelevinsky, B.A. Brown, N. Frazier, M. Horoi, Phys. Rep. 276 (1996) 315. [20] H.A. Weidenmüller, G.E. Mitchell, Rev. Mod. Phys. 81 (2009) 539. [21] B.A. Brown, B.H. Wildenthal, Annu. Rev. Nucl. Part. Sci. 38 (1988) 29. [22] T. Ericson, Adv. Phys. 9 (1960) 425. [23] V. Zelevinsky, Annu. Rev. Nucl. Part. Sci. 46 (1996) 237. [24] F. Borgonovi, F.M. Izrailev, L.F. Santos, V.G. Zelevinsky, Phys. Rep. 626 (2016) 1. [25] S. Karampagia, V. Zelevinsky, Phys. Rev. C 94 (2016) 014321. [26] P. Demetriou, S. Goriely, Nucl. Phys. A 695 (2001) 95. [27] S. Goriely, S. Hilaire, A.J. Koning, Phys. Rev. C 78 (2008) 064307. [28] M. Scott, M. Horoi, Europhys. Lett. 91 (2010) 52001. [29] A.V. Ignatyuk, K.K. Istekov, G.N. Smirenkin, Sov. J. Nucl. Phys. 29 (1979) 450. [30] M. Horoi, V. Zelevinsky, Phys. Rev. C 81 (2010) 034306.