Deformation effect on spectral statistics of nuclei

Available online at www.sciencedirect.com

ScienceDirect Nuclear Physics A 970 (2018) 29–43 www.elsevier.com/locate/nuclphysa

Deformation effect on spectral statistics of nuclei H. Sabri ∗ , A. Jalili Majarshin Department of Physics, University of Tabriz, Tabriz 51664, Iran Received 7 September 2017; received in revised form 26 October 2017; accepted 2 November 2017 Available online 8 November 2017

Abstract In this study, we tried to get significant relations between the spectral statistics of atomic nuclei and their different degrees of deformations. To this aim, the empirical energy levels of 109 even–even nuclei in the 22 ≤ A ≤ 196 mass region are classified as their experimental and calculated quadrupole, octupole, hexadecapole and hexacontatetrapole deformations values and analyzed by random matrix theory. Our results show an obvious relation between the regularity of nuclei and strong quadrupole, hexadecapole and hexacontatetrapole deformations and but for nuclei that their octupole deformations are nonzero, we have observed a GOE-like statistics. © 2017 Elsevier B.V. All rights reserved. Keywords: Deformed nuclei; Spectral statistics; Quadrupole, octupole, hexadecapole and hexacontatetrapole deformations, Nilsson perturbed-spheroid parameterization; Random matrix theory (RMT); Maximum Likelihood Estimation (MLE); Gaussian Orthogonal Ensemble (GOE)

1. Introduction The microscopic many-body interaction of particles in Fermi systems such as heavy nuclei is rather complicated. Several theoretical approaches to the description of the Hamiltonian which are based on the statistical properties of its discrete levels are applied for solutions of such realistic problems. For a quantitative measure for the degree of chaoticity in the many-body forces, the statistical distributions of the spacing between the nearest-neighboring levels were introduced in relation to the so-called Random Matrix Theory (RMT) [1–5]. The RMT and its different statis* Corresponding author.

E-mail address: [email protected] (H. Sabri). https://doi.org/10.1016/j.nuclphysa.2017.11.002 0375-9474/© 2017 Elsevier B.V. All rights reserved.

30

H. Sabri, A. Jalili Majarshin / Nuclear Physics A 970 (2018) 29–43

tical measures are commonly used to analyze the fluctuation properties of the quantum system’s spectra. In this model, a chaotic system is described by an ensemble of random matrices subject only to the symmetry restrictions [3–7], e.g. systems with time-reversal symmetry such as atomic nuclei are described by Gaussian Orthogonal Ensemble (GOE). On the contrary, integrable systems lead to level fluctuations that are well described by the Poisson distribution, i.e., levels behave as if they were uncorrelated [1–15]. In the recent studies [5–26], different statistical analyses have been accomplished on the nuclear system’s spectra to obtain statistically relevant samples. The spectral statistics of nuclei which were classified in the different mass regions have been described by Shriner et al. [4,25] or, the fluctuation properties of nuclei which were classified as their excitation energies ratios and type of quadrupole deformation, e.g. prolate or oblate, have been investigated by Abul-Magd et al. [5,14], are some of these comprehensive analyses. Gomez et al. [3,6] have considered the effect of shell model configurations on spectral statistics and Molina has established the relation between the pairing and spectral statistics [11]. Also, the effect of the temperature on the spectral statistics of nuclear systems has been studied in Refs. [23,26] and the effect of spin on nuclear spectral statistics and also the shell correction as a function of angular momentum was calculated by extending the thermodynamical method in related to rotating nuclei in Ref. [17], similar to what have done for nonrotating nuclei. The results of such descriptions suggest an obvious relation between the chaoticity of considered systems and some parameters such as mass, quadrupole deformation parameter, spin and etc. The effect of nuclear deformation on level statistics has been studied by Al-Sayed in Ref. [27] with emphasis on the only experimental values of quadrupole deformation parameter. The result of this study which considered the 31 nuclei in the rare-earth region show a non-trivial dependency between the nuclear spectral statistics and the quadrupole deformation and indicate the regularity in strongly deformed nuclei and especially in those having an oblate deformation. In the present study, we have developed similar consideration but with more nuclei and also other degrees of deformations which are expressed in the spherical-harmonic expansion together with the experimental values of quadrupole deformation [28–34]. We have focused on 2+ levels of even-mass nuclei for their relative abundances. Sequences are prepared by using all the available empirical data [28] which are classified as their deformation parameters and analyzed via Maximum Likelihood technique in the nearest neighbor spacing distribution (NNSD) framework. 2. Different degrees of deformations The experimental values of β2 , quadrupole deformation which is the most commonly used quantity in the description of deformed nuclei, are available in different lectures such as [29–34] which we have used them for one of our considered sequences. Al-Sayed has been analyzed the spectral statistics of limited numbers of nuclei, e.g. 31 nuclei, by qualification of this quantity. To achieve a sufficient number of energy levels within each interval, he had used the absolute value of β2 and compared the results by analyzing the data set in terms of the experimental β2 values, as deduced from the B(E2; 0+ → 2+ ) values [27]. He got a non-trivial dependency between the nuclear spectral statistics and the quadrupole deformation and indicate regularity in strongly deformed nuclei, β2 ≈ 0.300, and especially in those having an oblate deformation. We have tried to expand this study with considering the effect of high-order deformations, e.g. octupole, e.g. β3 (≡ ε3 ), hexadecapole, e.g. β4 (≡ ε4 ), and hexacontatetrapole deformations, e.g.

H. Sabri, A. Jalili Majarshin / Nuclear Physics A 970 (2018) 29–43

31

β6 (≡ ε6 ), on spectral statistics. The experimental values for these quantities are not available (or not suggested for all of our considered nuclei) and therefore, we must use the calculated values. These quantities, β3 , β4 and β6 , are taken from macroscopic–microscopic calculations [32–34]. In the first model of geometrical collective description [18], the nucleus was modeled as a charged liquid drop by Bohr and Mottelson. This model exhibits the moving of nuclear surface by an expansion in spherical harmonic with time-dependent shape parameters as coefficients:   ∞  l  R(θ, ϕ, t) = Rav 1 + αlμ (t)Ylμ (θ, ϕ) , (1) l=0 μ=−l

where R(θ, ϕ, t) represent the nuclear radius in the direction (θ, ϕ) at a time t and Rav is the radius of the spherical nucleus. Different l-values denote a translation or deformation seems at the collective excitation of the nucleus. The values of l and μ establish the surface coordinate as a function of θ and ϕ respectively. For axially symmetric nuclei, the nuclear radius can be rewritten as:   R(θ, ϕ) = Rav 1 + β2 Y20 (θ, ϕ) . (2) The quadrupole deformation parameter, β2 (= α20 ), can be related to the axes of the spheroid by:  4 π R β2 = . (3) 3 5 Rav Rav = R0 A1/3 is the average radius, R describes the difference between the semimajor and semiminor axes and consequently, the larger values of β2 explore the strongly deformed nuclei [18]. For other degrees of deformations, we have to use the calculated values which these are two different methods to express deformation: i) εl , the calculated ground-state deformations which are determined in the Nilsson perturbed-spheroid parameterization [30–32] which define the radius of nuclei as [32]:

  R0 2 2 r(θ, ϕ) = 1 − ε2 cos γ + π ω0 /ω˙ 0 3 3

−1/2  2 2 2 × 1 − ε2 cos γ − π 1 − ε2 cos γ 3 3 3 

1 2 2 1 2 × 1 − ε2 cos γ − ε2 cos γ + ε2 cos γ + ε2 cos 2γ u2 3 9 3 1/2

1/2

   1 2 − ε2 sin γ 1 − ε2 cos γ 1 − u2 u 3 3

1/2    2 1 2 1 × 1 − ε2 cos γ 3u − 1 + ε2 sin γ 1 − u2 v + 2ε1 P1 (u) 3 2 3 −1/2 + 2ε3 P3 (u) + 2ε4 V4 (u, v) + 2ε5 P5 (u) + 2ε6 P6 (u) . (4) (Pl is the Legendre Polynomials) or ii) the method which expresses the calculated deformations of the nuclear ground-state, e.g. βl in a spherical-harmonics expansion, as:

32

H. Sabri, A. Jalili Majarshin / Nuclear Physics A 970 (2018) 29–43

βlm =

√

 4π 

r(θ, ϕ)Ylm (θ, ϕ)dΩ r(θ, ϕ)Y00 (θ, ϕ)dΩ

,

(5)

[30,31]. A detailed description of these methods and the parameters which are used in the Equations (4)–(5) are available in Refs. [30–34]. In this study, we have used the calculated values and therefore, interested readers are suggested to consider these references. Also, we have considered the values of both models but any obvious difference doesn’t observe and consequently, we have used the β2 , β3 , β4 and β6 which are derived via Eq. (5) in our classifications. We have qualified nuclei as the deformed according to the liquid drop model calculation by P. Moller et al. [30,31], therefore, the strongly deformed nuclei have the great values of deformation parameters. As have explained by Shriner et al. in Ref. [25], the spacing distributions can distinguish between GOE and Poisson behavior with sequences of as few as 25 levels. Now if we combine several such sequences, the NNS distribution seems to be more reliable in our statistics. In order to prepare sequences by different nuclei with the available empirical data taken from Refs. [28,29], we have followed the same method given in Ref. [25]. Namely, we have considered nuclei in which the spin-parity J π assignments of at least five consecutive levels are definite. In cases where the spin-parity assignments are uncertain and where the most probable value appeared in brackets, we admit this value. We terminate the sequence for each nucleus when we reach a level with unassigned J π . We focus on 2+ levels for even mass nuclei for their relative abundance in the considered nuclei. In this approach, we have achieved 109 nuclei as presented in Table 1. These nuclei have, at least, one non-zero deformation values. 3. Method of analysis Random matrix theory allows us to establish a connection between the statistical properties of energy spectra and quantum chaos. The work of Berry and Tabor [35], which shows that integrable systems lead to energy-level fluctuations that are well described by the Poisson distribution, and the work of Bohigas, Giannoni, and Schmit [36], which conjectured that spectral fluctuation properties of chaotic systems are well described by random matrix theory (known as the BGS conjecture, later proved by Heusler et al. [37]), can be considered as a definition of quantum chaos in terms matrix theory came from nuclear physics. The fluctuation properties of nuclear spectra have been considered by different statistics such as Nearest Neighbor Spacing Distribution (NNSD) [1–4] and the Dyson–Mehta 3 statistic [3–7]. Nearest Neighbor Spacing Distribution (NNSD) is the most commonly used statistics which exhibit the statistical situation of considered systems as compared with different limits of Random matrix theory (RMT). The NNSD statistics requires complete and pure level scheme. The information on regular and chaotic nuclear motion available from experimental data is rather limited because the analysis of energy levels requires the knowledge of sufficiently large pure sequences, i.e., consecutive levels sample all with the same quantum numbers (J, π ) in a given nucleus. This means one needs to combine different level schemes to prepare the sequences and perform a significant statistical study. Such analyses [20–27] showed that the NNS distributions are intermediate between Poisson (order) and chaotic (Gaussian unitary ensemble (GOE)) limits. Different distribution functions [12–15] have been proposed to describe the spectral statistics of considered sequences quantitatively in comparison with different limits of RMT. To describe the statistical properties of nuclei in different categories via NNS distributions, we must have sequences of unit mean level spacing, similar to every statistical analysis which using the RMT predictions. This requirement is equivalent to using levels with the same total quantum number (J ) and same parity. For a given spectrum {Ei }, it is necessary to separate it into the fluctuation part and the smoothed average part, whose

H. Sabri, A. Jalili Majarshin / Nuclear Physics A 970 (2018) 29–43

33

Table 1 Deformed nuclei which have at least one non-zero deformation parameters. N denotes the number of levels, (β2 )exp , (β2 )Th. , β3 , β4 , β6 are respectively, the experimental and theoretical quadrupole, octupole, hexadecapole and hexacontatetrapole deformations parameters and T1/2 explores the half-life of the selected nucleus. Also, the energy interval of each nucleus is expressed in table. Nuclei

R4/2

T1/2

N

(β2 )exp

(β2 )Th.

β3

β4

β6

22 Ne

2.634 2.652 3.012 2.387 2.137 2.595 2.361 1.999 2.203 2.186 2.240 2.467 1.980 2.625 1.805 1.973 4.397 2.266 2.470 2.401 1.652 2.462 2.561 2.362 1.691 1.991 2.291 2.326 2.358 2.232 2.071 2.535 1.898 2.553 2.650 1.458 2.440 2.376 1.437 2.316 2.564 2.199 3.005 1.600 1.917 2.121

STABLE 3.875s STABLE STABLE 2.22s STABLE STABLE STABLE STABLE 170.3m STABLE STABLE STABLE 440ms STABLE STABLE 13.9s 60.0Y 21.56h >1.3E+18Y STABLE STABLE STABLE 2.0s STABLE STABLE 9.186h ≥7.0E20Y STABLE 270.95d STABLE 88.0m 8.40d STABLE STABLE 3.26m 14.8h STABLE STABLE 25.34d 75.3s 1.07s 0.653s STABLE STABLE 7.3E+18Y

5 8 6 11 7 5 8 6 7 7 13 24 5 6 23 6 5 5 5 9 12 7 14 8 6 6 11 11 6 8 5 5 6 7 10 5 6 9 7 5 5 5 6 5 21 6

0.566 0.670 0.613 0.484 0.438 0.412 0.330 0.314 0.247 0.245 0.253 0.161 0.266 0.121 0.245 0.262 0.065 0.260 0.368 0.290 0.212 0.250 0.262 – 0. 179 0.197 0.216 0.236 0.221 0.193 0.226 0.239 0.227 0.232 0.192 0.143 0.290 0.153 0.134 0.290 0.117 0.150 0.409 0.101 0.168 0.162

0.326 0.326 0.374 −0.310 −0.353 −0.478 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.199 −0.087 0.000 −0.096 0.209 0.219 −0.215 −0.275 −0.241 0.153 −0.283 0.153 0.154 0.053 0.400 0.062 0.053 0.053 0.255 0.338 0.357 0.053 0.180 0.244

– – – – – – – – – – – – – – – – −0.015 – – −0.024 −0.023 −0.028 – – −0.024 – 0.003 – – – – – – – – – – – – – – – – – – –

0.225 0.225 −0.053 0.186 0.226 0.250 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 −0.019 −0.020 0.000 0.012 −0.031 −0.031 0.005 0.003 −0.014 0.000 −0.011 −0.024 −0.049 −0.007 −0.024 −0.007 −0.007 0.001 0.001 0.041 0.056 0.001 0.022 0.023

0.011 1274 6819 0.011 1247 5838 −0.010 1368 9284 −0.013 1808 7840 −0.017 1797 5229 −0.020 1779 8258 0.000 2235 7623 0.000 2230 7115 0.000 2127 6428 0.000 1292 5278 0.000 1970 7178 0.000 2167 8391 0.000 1460 3918 0.000 2213 5264 0.000 1524 5875 0.000 1157 3776 0.000 1026 4870 0.000 1083 4115 0.000 752 4640 0.000 783 4192 0.000 1434 5097 0.000 846 3744 −0.001 810 4348 −0.004 746 4226 0.000 1454 3273 0.004 1172 3269 −0.031 953 4330 −0.008 991 3538 0.038 1039 3212 0.062 1015 3400 0.027 1039 2944 −0.008 619 2319 0.048 862 2293 −0.024 666 2774 −0.049 654 4566 0.000 1454 3024 0.025 423 2570 0.001 881 3183 0.000 1564 4194 0.001 573 2885 −0.017 836 2710 0.014 814 2083 0.007 144 1681 0.000 934 3830 −0.020 787 3152 −0.015 535 2042 (continued on next page)

22 Mg 24 Mg 26 Mg 26 Si 28 Si 30 Si 32 S 34 S 38 S 36 Ar 38 Ar 40 Ar 38 Ca 42 Ca 44 Ca 50 Ca 44 Ti 48 Cr 50 Cr 52 Cr 56 Fe 58 Fe 64 Fe 58 Ni 62 Ni 62 Zn 64 Zn 66 Zn 68 Ge 70 Ge 78 Ge 72 Se 80 Se 82 Se 84 Se 76 Kr 84 Kr 86 Kr 82 Sr 94 Sr 96 Sr 98 Sr 92 Zr 98 Mo 100 Mo

Emin

Emax

34

H. Sabri, A. Jalili Majarshin / Nuclear Physics A 970 (2018) 29–43

Table 1 (continued) Nuclei

R4/2

T1/2

N

(β2 )exp

(β2 )Th.

β3

β4

β6

Emin

Emax

102 Mo

2.507 1.823 2.142 2.755 2.127 2.533 2.361 2.383 2.344 2.292 2.298 2.374 1.812 1.788 1.848 2.094 2.071 1.697 2.402 2.467 2.423 2.332 2.157 1.290 1.821 2.777 2.427 1.322 1.876 2.836 2.379 2.361 2.347 3.009 1.805 2.194 3.014 2.233 2.933 3.270 3.293 2.743 3.230 3.309 2.923 3.126 3.292 3.268 3.290

11.3m STABLE STABLE 11.6s 3.63d 21.03h >3.6E+20Y >1.9E+18Y STABLE STABLE >2,1E18Y 3.3E+19Y 4.11h >1.3E+21Y STABLE STABLE STABLE 17.63s 3.8m 40m STABLE STABLE STABLE >2.4E+21Y 14.08m 100m >3.0E+21Y STABLE 12.75d 2.22s >0.7E+14Y 284.91d 14.82m STABLE 70.9Y 1.08E14Y STABLE 3.0E+6Y STABLE STABLE STABLE 2.29h STABLE STABLE 18.87m 75.8m STABLE 2.0E+15Y STABLE

5 12 5 18 7 24 25 5 7 53 8 6 16 5 20 9 5 13 5 10 5 7 9 7 5 6 9 6 8 6 8 9 6 14 7 7 7 5 6 5 6 13 6 6 6 7 5 14 5

0.311 0.154 0.205 0.293 – 0.218 0.168 0.170 0.172 0.182 0.130 0.135 0.120 0.123 0.112 0.131 0.170 – 0.190 0.208 0.188 0.202 0.141 0.091 – 0.273 0.185 0.093 0.116 0.219 0.170 0.162 – 0.308 0.077 0.212 0.310 0.235 0.294 0.334 0.341 0.264 0.332 0.336 0.263 0.296 0.321 0.272 0.279

0.329 0.053 0.115 −0.250 0.088 −0.241 0.126 0.135 0.144 0.144 0.163 −0.241 0.027 0.018 0.000 −0.139 −0.113 0.000 0.244 0.245 0.170 0.143 0.000 0.000 0.000 0.256 0.143 0.000 0.000 0.199 0.107 0.144 −0.148 0.243 0.000 0.207 0.243 0.207 0.235 0.272 0.281 0.216 0.272 0.296 0.225 0.264 0.295 0.285 0.278

– – – – – – – – – – – – – – – – – – – – – – – – – – – – – −0.107 – −0.082 – – – – – – – – – – – – – – – – –

0.050 0.009 0.038 −0.044 0.019 −0.038 −0.002 −0.018 −0.033 −0.033 −0.040 −0.038 −0.008 −0.015 −0.008 −0.001 −0.010 0.000 0.031 0.006 0.002 0.008 0.000 0.000 0.000 −0.025 −0.008 0.000 0.000 0.100 −0.011 0.066 −0.030 0.090 0.000 0.050 0.073 0.041 0.046 0.053 0.040 0.034 0.037 −0.023 0.019 0.010 −0.025 −0.035 −0.080

−0.007 0.000 −0.002 0.010 −0.003 0.008 −0.007 −0.003 0.005 0.009 0.012 0.014 0.001 −0.001 0.000 0.008 0.000 0.000 −0.007 −0.006 −0.003 −0.002 0.000 0.001 0.002 −0.009 −0.005 0.000 0.000 0.008 0.000 0.002 −0.008 −0.007 0.000 −0.008 −0.006 −0.008 −0.005 −0.008 −0.013 −0.006 −0.006 −0.031 −0.005 −0.005 −0.020 −0.015 −0.004

269 832. 652 240 665 348 632 632 657 617 558 513 1212 1256 1293 564 602 606 337 322 388 442 667 1313 588 256 464 1435 602 181 552 397 530 121 784 344 123 334 137 86 80 192 102 78 166 123 84 90 93

1608 3261 2276 2367 3235 3013 3426 2486 2365 3428 2317 2118 3446 2721 3850 2508 2182 3714 1838 2050 2086 2361 2714 2979 2114 1810 2980 2794 2521 1968 2942 2405 2482 1958 2700 1470 1775 1507 1514 1556 1895 1697 1500 1416 1337 1512 1534 2529 1513

96 Ru 98 Ru 110 Ru 100 Pd 112 Pd 106 Cd 108 Cd 110 Cd 112 Cd 114 Cd 116 Cd 110 Sn 112 Sn 116 Sn 122 Te 124 Te 136 Te 118 Xe 120 Xe 126 Xe 128 Xe 132 Xe 136 Xe 138 Xe 126 Ba 132 Ba 138 Ba 140 Ba 146 Ba 136 Ce 144 Ce 140 Sm 152 Sm 148 Gd 152 Gd 154 Gd 154 Dy 156 Dy 160 Dy 162 Dy 158 Er 162 Er 170 Er 162 Yb 164 Yb 170 Yb 174 Hf 178 Hf

H. Sabri, A. Jalili Majarshin / Nuclear Physics A 970 (2018) 29–43

35

Table 1 (continued) Nuclei

R4/2

T1/2

N

(β2 )exp

(β2 )Th.

β3

β4

β6

Emin

Emax

184 W

3.273 3.233 3.154 2.934 2.819 2.680 2.707 2.674 2.491 2.478 2.465 2.502 2.501 2.491 2.546 1.656

STABLE >2.3E+19Y 21.84Y STABLE STABLE 56s 2.67m 17.3m 6.5E+11Y STABLE STABLE 20.0m 4.85h STABLE STABLE 37m

7 7 8 11 20 5 5 5 10 7 21 5 7 13 9 5

0.234 0.226 0.236 0.177 0.164 0.252 0.217 0.229 0.152 0.154 0.129 – – 0.114 0.106 –

0.240 0.230 0.239 0.164 0.155 0.265 0.255 0.247 −0.156 −0.156 −0.139 −0.130 −0.130 −0.122 −0.122 0.000

– – – – – – – – – – – – – – – –

−0.095 −0.107 −0.062 −0.080 −0.081 −0.007 −0.026 −0.044 −0.022 −0.029 −0.030 −0.032 −0.032 −0.032 −0.032 −0.008

0.010 0.020 −0.001 0.018 0.020 −0.007 −0.002 0.000 0.006 0.006 0.013 0.000 0.001 0.003 0.004 0.000

111 122 126 186 205 153 154 162 295 316 355 416 422 425 411 1049

1614 1322 1768 1708 1951 1351 1311 1611 1628 1793 2093 1850 2276 2012 1902 2124

186 W 182 Os 190 Os 192 Os 180 Pt 182 Pt 184 Pt 190 Pt 192 Pt 196 Pt 190 Hg 192 Hg 196 Hg 198 Hg 196 Pb

behavior is non-universal and cannot be described by RMT [1]. To do so, we include the number of the levels below E and write it as [4]: E N (E) =

ρ(E)dE = e(

E−E0 T )

E0

− e − T + N0 ,

(6)

0

where N0 establishes the number of levels with energies less than zero and must be assumed as zero. The best fit to N (E) (≡ F (E)) would be carried while a correct set of energies is prepared by means of, Ei = Emin +

F (Ei ) − F (Emin ) (Emax − Emin ). F (Emax ) − F (Emin )

(7)

Both Emax and Emin remain unchanged with this transformation. These transformed energies should now display on average a constant level density. This unfolded level sequence, {Ei }, is dimensionless and has a constant average spacing of 1 but actual spacing exhibits frequently

) − (E ). Distribution strong fluctuation. Nearest neighbor level spacing is defined as si = (Ei+1 i P (s) will be in such a way in which P (s)ds is the probability for the si to lie within the infinitesimal interval [s, s + ds]. The NNS probability distribution function of nuclear systems which spectral spacing follows the Gaussian Orthogonal Ensemble (GOE) statistics is given by Wigner distribution [1]: πs 2 1 P (s) = πse− 4 . (8) 2 This distribution exhibits the chaotic properties of spectra. On the other hand, the NNSD of systems with regular dynamics is generically represented by Poisson distribution:

P (s) = e−s .

(9)

It is well known that the real and complex systems such as nuclei are usually not fully ergodic and neither are they integrable [38–55]. Different distribution functions have been proposed to com-

36

H. Sabri, A. Jalili Majarshin / Nuclear Physics A 970 (2018) 29–43

pare the spectral statistics of considered systems with regular and chaotic limits quantitatively and also explore the interpolation between these limits [12–15]. Berry–Robnik distribution [13] is one of the popular distributions:

 1 1 P (s, q) = q + π(1 − q)s × exp −qs − π(1 − q)s 2 . (10) 2 4 This distribution is derived by assuming the energy level spectrum is a product of the superposition of independent subspectra which are contributed respectively from localized eigenfunctions into invariant (disjoint) phase space and interpolates between the Poisson and Wigner with q = 1 and 0, respectively. To consider the spectral statistics of sequences, one must compare the histogram of sequence with Berry–Robnik distribution and extract its parameter via estimation techniques. To avoid the disadvantages of estimation methods such as least square fitting (LSF) technique which has some unusual uncertainties for estimated values and also exhibit more approaches to chaotic dynamics, Maximum Likelihood (ML) technique has been used [10] which yields very exact results with low uncertainties in comparison with other estimation methods. The MLE estimation procedure has been described in detail in Refs. [10,15]. Here, we outline the basic ansatz and summarize the results. In order to estimate the parameter of distribution, the Likelihood function is considered as a product of all P (s) functions: n n   1 1 2 L(q) = P (si ) = (11) q + π(1 − q)si e−qsi − 4 π(1−q)si . 2 i=1

i=1

The desired estimator is obtained by maximizing the likelihood function, Eq. (11),   

1 − πs2 i πsi2 f: − s − . i q + π2 (1 − q)si 4

(12)

We can estimate “q” by high accuracy via solving the above equation by the Newton–Raphson method: F (qold ) qnew = qold −

, F (qold ) which is terminated to the following result:  qnew = qold −

πs

1− 2 i πs qold + 2 i (1−qold )



+ πs



si +

−(1− 2 i )2 (qold + 12 π(1−qold )si )2

πsi2 4

.

(13)

In the ML-based technique, we have followed the prescription was explained in Ref. [10], namely maximum likelihood estimated parameters correspond to the converging values of iterations Eq. (13) which for the initial values we have chosen the values of parameters were obtained by LSF method. 4. Results The search for a phenomenological ‘control parameter’ for describing the evolution of the stochastic nature of nuclear dynamics became an area of nuclear structure research in the last two decades. In the present work, we examine the use of the quadrupole, octupole, hexadecapole and hexacontatetrapole deformations parameters as a probe of nuclear structure. In spite of the

H. Sabri, A. Jalili Majarshin / Nuclear Physics A 970 (2018) 29–43

37

Fig. 1. A detailed description of chaoticity degrees in different mass regions. The most regular dynamics are observed for the heaviest nuclei.

absence of a complete theoretical study defining certain fixed values of β2 , β3 , β4 and β6 , we tend to classify nuclei into fixed intervals of these deformation degrees to allow a qualitative study. We recall that the analysis of many short sequences of levels tends to overestimate the degree of chaoticity measured by parameter q. We focus our attention not only on the absolute values of q but also on the way q changes with different degrees of deformation. To prepare sequences and consider the statistical situation of them, we tend to classify the considered even–even nuclei in different categories which have at least five consecutive levels with the definite spin-parity J π assignment. These sequences are unfolded and then analyzed via Berry–Robnik distribution and Maximum Likelihood estimation technique. Since, the exploration of the majority of short sequences yields an overestimation about the degree of chaotic dynamics which are measured by distribution parameters, i.e. q, therefore, we would not concentrate only on the implicit values of these quantities and examine a comparison between the amounts of this quantity in any of tables. The ML-based predictions for the chaoticity (or regularity) degrees of all of the considered nuclei are presented in Fig. 1. To get the relation of chaoticity and mass of nuclei, we have classified all of the considered nuclei in different sequences with at least 50 spacing. The results present a detailed description of the statistical situation in different mass regions and confirm previous predictions [4,10]. The most chaoticity observed in the light nuclei while we go to heavier ones, the regular behavior is dominant. In the second analysis, we have used the half-lives of considered nuclei as the measure for the classification. As have presented in Table 2, we have observed the most regular dynamics for stable nuclei and also, when the half-lives are decreased, a transition from regularity to chaotic behavior is observed in sequences. We have tried to get a meaningful relation between the different deformed categories and the chaoticity degrees in considered sequences. To this aim, we have classified the nuclei as their non-zero deformation values in different sequences.

38

H. Sabri, A. Jalili Majarshin / Nuclear Physics A 970 (2018) 29–43

Table 2 The chaoticity parameters, “q” Berry–Robnik distribution parameter, are determined for different sequences which are classified as their half-lives. n is the number of nuclei in each category and N presents the number of spacing. Sequence

n

N

q

Stable nuclei T1/2 ≈ year (s) T1/2 ≈ day (s) T1/2 ≈ hour (s) T1/2 ≈ minute (s) T1/2 ≈ second (s)

54 19 6 6 16 12

606 164 40 92 100 92

0.91 ± 0.12 0.82 ± 0.09 0.70 ± 0.10 0.58 ± 0.07 0.42 ± 0.08 0.33 ± 0.05

Fig. 2. The variation of chaoticity measure (“q”, the parameter of Berry–Robnik distribution) versus the variation of experimental quadrupole deformation. The regular behavior of strongly deformed nuclei is obvious.

i) Effect of experimental quadrupole deformation on spectral statistics We have used the available values of experimental quadrupole deformation for the considered nuclei and classified them in the 0.077 ≤ β2 ≤ 0.670 interval. Nuclei are grouped in sets which have at least 50 spacing. The sequences are unfolded and then analyzed via Berry–Robnik distribution. Results are presented in Fig. 2. Similar to the predictions of Al-Sayed in Ref. [27], we observed a regular behavior for strongly deformed nuclei. Except some trivial fluctuations, a direct relation is apparent between the deformation and the regularity. The apparent regularity for these deformed nuclei confirm the predictions of GOE limit which suggest more regular dynamics for deformed nuclei in comparison with the spherical nuclei, e.g. magic or semi-magic nuclei. One can expect the spherical nuclei which have shell model spectra explore predominantly less regular dynamics in comparison with the deformed ones. This result is known as Abul-Magd– Weidenmuller chaoticity effect [5] where suggest the suppression of chaotic dynamics due to the rotation of nuclei.

H. Sabri, A. Jalili Majarshin / Nuclear Physics A 970 (2018) 29–43

39

Fig. 3. Similar to Fig. 2, we have used the calculated quadrupole deformation for our classification. The regularity of deformed nuclei and also the more regular behavior of nuclei with negative β2 are appeared.

ii) Effect of the theoretical quadrupole deformation on spectral statistics In Fig. 3, we have presented the chaoticity degrees of different sequences which are classified as their theoretical quadrupole deformation values. With regard to the negative and positive values of this quantity, we have considered the −0.478 ≤ β2 ≤ 0.400 interval for our analysis. Results show a partly symmetric behavior about the β2 = 0, e.g. correspond to spherical nuclei and describe more regular dynamic for sequences with β2 = 0. The spherical (magic or semimagic) nuclei which are expected to have shell model spectra, and thus agree with GOE. Also, our results suggest more regularity for systems which their calculated β2 values are negative. An observed minimum of chaoticity is centered at β2 ≈ −0.1 and 0.25. Although the statistical errors are not good enough for drawing a conclusion, this may indicate that something interesting is happening through that interval. iii) Effect of theoretical octupole deformation on spectral statistics As have presented in Table 1, the calculated values of octupole deformation are available for the limited number of nuclei, 8 nuclei and 58 spacing, and this makes impossible to classify nuclei in different sequences. These conditions force us to consider two sequences, a) nuclei which their β3 are non-zero (except one nucleus, others have negative values) and b) nuclei with β3 = 0. The ML-based values for Berry–Robnik distribution are q = 0.12 ± 0.09 and q = 0.83 ± 0.06 for the sequences (a) and (b), respectively. These results suggest the chaotic behavior for nuclei with non-zero octupole degrees of deformation. This result in combination with our latest investigation about octupole degrees of deformation [56], suggests the role of protons in unclosed shells in this type of deformation and chaotic behavior but for general conclusion, we need more data which may available in odd-mass nuclei and we would consider them in future studies.

40

H. Sabri, A. Jalili Majarshin / Nuclear Physics A 970 (2018) 29–43

Fig. 4. We presented the variation of a chaoticity parameter as a function of calculated hexadecapole deformation and the average number of neutrons in each sequence. A regular behavior is apparent for systems with negative β4 values and great N ave number.

iv) Effect of the theoretical hexadecapole deformations on spectral statistics In this subsection, we tried to classify nuclei as their non-zero calculated hexadecapole deformations. We have divided nuclei in different sequences such they have at least 50 spacing in the −0.107 ≤ β4 ≤ 0.250 interval. The results for the chaoticity degrees of these sequences versus their values β4 and also the average number of neutrons (for nuclei in each sequence) are presented in Fig. 4. We have selected this manner to get a meaning about the relation of the neutron (or proton) number and chaoticity. Similar to results of subsection (ii), we have observed more regular dynamics for sequences with β4 = 0. Two remarkable minima are nearly centered at β4 ≈ −0.057 and 0.103; these two regions have nearly the same chaoticity parameter that may relate to well-deformed nuclei. The great numbers of neutrons and protons (outside of closed shells) of such regular nuclei may be related to the proton–proton or neutron–neutron interaction. As have mentioned in Refs. [44,45], the relatively weak strength of the only neutron– neutron (or proton–proton) interaction is unable to destroy the regular single-particle mean-field motion completely. In some nuclei with both protons and neutrons in valence orbits, on the other hand, the strong proton–neutron interaction would appear to be sufficient to destroy the regular mean-field motion. v) Effect of the theoretical hexacontatetrapole deformations on spectral statistics We have used the calculated hexacontatetrapole deformations, β6 , to consider the evolution of chaoticity degrees for different sequences. We have classified the −0.049 ≤ β6 ≤ 0.048 interval to different sequences which have at least 40 spacing (due to the zero values of β6 for the majority of considered nuclei). The regular behavior, as have presented in Fig. 5, is obvious for nuclei with non-zero β6 values, similar to what have done for other deformation degrees. In this figure similar to what has done in Fig. 4, we considered the variation of chaoticity degrees versus β6 and average neutron numbers of sequences. Also, nuclei which have negative β6 values describe more

H. Sabri, A. Jalili Majarshin / Nuclear Physics A 970 (2018) 29–43

41

Fig. 5. Similar to Fig. 4, the variation of a chaoticity parameter as function of calculated hexacontatetrapole deformation and the average number of neutrons in each sequence. A regular behavior is apparent for systems with negative β6 values and great N ave number.

regular dynamics. The results show the minimum chaoticity in three regions which correspond to β6 ≈ −0.030, −0.008 and 0.024 and nuclei which have extra neutrons in comparison with their proton number. This regularity may be related to the strength of pairing force in comparison with Coulomb force but for a significant conclusion, we need to consider more general cases. From these tables and figures one can conclude that, the deformed nuclei show more regular behavior in comparison with spherical ones. These results are verified by other classifications which are occurred via high order deformations. Also, nuclei which have negative values of calculated deformation parameters show more regular behavior. For classification which has done by experimental β2 values, this result may be interpreted as follows: the degree of interaction between single-particle motion which is chaotic and collective motion of whole nucleons which is believed to be more regular is weaker in the case of oblate deformed nuclei than for prolate ones. 5. Summary and conclusion We have investigated the spectral statistics of 109 even–even nuclei by using all the available experimental data. Berry–Robnik distribution and MLE technique have been employed to consider the statistical situation of different sequences. We have classified nuclei as their mass, half-lives and experimental and calculated values of different degrees of deformations. The difference in the chaoticity parameter of each sequence is statistically significant. We have found an obvious relation between the longest half-lives and the regularity of different sequences. Also, the regular dynamic is dominant for well-deformed nuclei in comparison with other ones. This result is verified by other classification which have done by using other degrees of deformations. Also, more regular behavior are suggested to nuclei which have negative values of calculated deformation parameters. These results may yield deep insight into the single-particle motion in the mean field formed by the deformed systems.

42

H. Sabri, A. Jalili Majarshin / Nuclear Physics A 970 (2018) 29–43

Acknowledgement This work is published as a part of research project supported by the University of Tabriz Research Affairs Office. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47]

H.A. Weidenmuller, G.E. Mitchel, Rev. Mod. Phys. 81 (2009) 539. T.A. Brody, J. Flores, J.P. French, P.A. Mello, A. Pandey, S.S.M. Wong, Rev. Mod. Phys. 53 (1981) 385. J.M.G. Gómez, et al., Phys. Rep. 499 (2011) 103. J.F. Shriner, G.E. Mitchell, T. von Egidy, Z. Phys. A 338 (1991) 309. A.Y. Abul-Magd, et al., Ann. Phys. (N.Y.) 321 (2006) 560. A. Relano, J.M.G. Gomez, R.A. Molina, J. Retamosa, E. Faleiro, Phys. Rev. Lett. 89 (2002) 244102. Y. Alhassid, Rev. Mod. Phys. 72 (2000) 895. Declan Mulhall, Phys. Rev. C 83 (2011) 054321. T. von Egidy, A.N. Behkami, H.H. Schmidt, Nucl. Phys. A 481 (1988) 189. M.A. Jafarizadeh, et al., Nucl. Phys. A 890–891 (2012) 29. R.A. Molina, Eur. Phys. J. A 28 (2006) 125. T.A. Brody, Lett. Nuovo Cimento 7 (1973) 482. M.V. Berry, M. Robnik, J. Phys. A 17 (1984) 2413. A.Y. Abul-Magd, A. Al-Sayed, Phys. Rev. C 74 (2006) 037301. H. Sabri, et al., Random Matrices: Theory Appl. 3 (2014) 1450017. M. Rajasekaran, T.R. Rajasekaran, N. Arunachalam, Phys. Rev. C 37 (1988) 307. V.S. Ramamurthy, S.S. Kapoor, S.K. Kataria, Phys. Rev. Lett. 25 (1970) 386. A. Bohr, B.R. Mottelson, Nuclear Structure, vol. II, W. A. Benjamin, Inc., 1971. M. Spieker, S. Pascu, A. Zilges, F. Iachello, Phys. Rev. Lett. 114 (2015) 192504. V.V. Flambaum, F.M. Izrailev, Phys. Rev. E 56 (1997) 5144. E.K. Warburton, B.A. Brown, Phys. Rev. C 46 (1992) 923. M. Horoi, V. Zelevinsky, B.A. Brown, Phys. Rev. Lett. 74 (1995) 5194. T. von Egidy, A.N. Behkami, H.H. Schmidt, Nucl. Phys. A 481 (1988) 189. Y. Alhassid, G.F. Bertsch, S. Liu, H. Nakada, Phys. Rev. Lett. 84 (2000) 4313. J.F. Shriner Jr., E.G. Bilpuch, G.E. Mitchell, Z. Phys. A 332 (1989) 45. Y. Alhassid, G.F. Bertsch, L. Fang, Phys. Rev. C 68 (2003) 044322. A. Al-Sayed, J. Stat. Mech. (2009) P02062. National Nuclear Data Center (Brookhaven National laboratory), chart of nuclides, http://www.nndc.bnl.gov/ chart/reColor.jsp?newColor=dm. Live chart, table of nuclides, http://www-nds.iaea.org/relnsd/vcharthtml/VChartHTML.html. P. Moller, et al., Nucl. Data Sheets 59 (1995) 185. P. Moller, et al., Nucl. Data Sheets (2017), https://doi.org/10.1016/j.adt.2015.10.002. P. Moller, et al., Phys. Rev. Lett. 103 (2009) 212501. S.E. Larsson, I. Ragnarsson, S.G. Nilsson, Phys. Lett. B 38 (1972) 269. S.E. Larsson, Phys. Scr. 8 (1973) 17. M.V. Berry, M. Tabor, Proc. R. Soc. Lond. A 356 (1977) 375. O. Bohigas, M.J. Giannoni, C. Schmit, Phys. Rev. Lett. 52 (1984) 1. S. Heusler, S. Müller, A. Altland, P. Braun, F. Haake, Phys. Rev. Lett. 98 (2007) 044103. R. Schafer, et al., Phys. Rev. E 66 (2002) 016202. A. Relaño, J. Retamosa, E. Faleiro, J.M.G. Gómez, Phys. Rev. E 72 (2005) 066219. P. Cejnar, P. Stránský, AIP Conf. Proc. 1575 (2014) 23. J.M.G. Gómez, et al., Phys. Lett. B 480 (2000) 245. K. Langanke, D.J. Dean, P.B. Radha, Y. Alhassid, S.E. Koonin, Phys. Rev. C 52 (1995) 718. V. Paar, D. Vorkapic, Phys. Lett. B 205 (1988) 7. V. Paar, D. Vorkapic, Phys. Rev. C 41 (1990) 2397. Y. Alhassid, D. Vretenar, Phys. Rev. C 46 (1992) 1334. Y. Alhassid, R.D. Levine, Phys. Rev. A 40 (1989) 5277. A. Hamoudi, R.G. Nazmitdinov, E. Shahallev, Y. Alhassid, Phys. Rev. C 65 (2002) 064311.

H. Sabri, A. Jalili Majarshin / Nuclear Physics A 970 (2018) 29–43

[48] [49] [50] [51] [52] [53] [54] [55] [56]

J.M.G. Gómez, K. Kar, V.K.B. Kota, R.A. Molina, J. Retamosa, Phys. Lett. B 567 (2003) 251. J.M.G. Gómez, et al., J. Phys. Conf. Ser. 580 (2015) 012045. J.M.G. Gomez, et al., J. Phys. Conf. Ser. 267 (2011) 012061. V.E. Bunakov, Phys. At. Nucl. 77 (2014) 1550. H. Sabri, Nucl. Phys. A 941 (2015) 364. D.C. Meredith, S.E. Koonin, M.R. Zirnbauer, Phys. Rev. A 37 (1988) 3499. Xi-Zhen Wu, Zhu-Xia Li, Ning Wang, J.A. Maruhn, Commun. Theor. Phys. 39 (2003) 597. V.R. Manfredi, J.M.G. Gomez, L. Salasnich, arXiv:nucl-th/9805049. A. Jalili Majarshin, M.A. Jafarizadeh, H. Sabri, Eur. Phys. J. Plus 132 (2017) 418.

Further reading [57] H. Sabri, R. Malekzadeh, Nucl. Phys. A 963 (2017) 78.

43