International Journal of Atomic and Nuclear Physics
(ISSN: 2631-5017)
Volume 4, Issue 1
Research Article
DOI: 10.35840/2631-5017/2514
Nuclear Poincaré Cycle Deduced From Regularities in Neutron Resonances in 40Ca and 56Fe - A Comparison with the Semi-Classical Model
Makio Ohkubo*
Table of Content
Figures
Figure 1: Time evolution of the CN at resonance...
Time evolution of the CN at resonance. (a) A toy model that shows the recurrence of many time-quantized oscillators rotating with arbitrary frequencies. (b) Penetrating component, which excites the CN and recurs (flare-up on the CN surface) at coalescent phases with the Poincaré periodicity, synchronizing with the passing component (C) at resonance.
Figure 2: (Sn/LCE) spectra of 40Ca+n s-wave resonances...
(Sn/LCE) spectra of 40Ca+n s-wave resonances. The pairs of resonances that form a peak at x = 14.51 are described in the text.
Figure 3: Repulsion between U238 and alpha particle...
vs. neutron energy of 40Ca+n s-wave resonances. Resonances denoted as hearts are members of the family, with their energies and ratios to the nodal resonance at 576 keV indicated. The inset shows the doorway coupled to the nodal resonance composed of 3 oscillators of time periods 6, 20 and 60 with LCM = 60 .
Figure 4: (Sn/LCE) spectra of 56Fe + n s-wave...
(Sn/LCE) spectra of 56Fe + n s-wave resonances. The pairs of resonances that form a peak at x = 21.78 are described in the text.
Figure 5: vs. neutron energy of 56Fe + n s-wave...
vs neutron energy of 56Fe + n s-wave resonances. Resonances denoted as diamonds are members of the family, with their energies and ratios to the nodal resonance at 351 keV indicated. In the inset, are shown the doorway coupled to the nodal resonance composed of two oscillators of time periods 5 and 100 with LCM=100 .
Tables
Table 1: 40Ca + n S-wave resonance parameters below 1400 keV [12,13]. Observed resonance energies Enobs, recoil corrected resonance energies En, and neutron widths Γn are shown.
Table 2: 40Ca + n pairs of s-wave resonances that form the 14.51 peak in Figure 2. E2/E1 = b/a, (a, b < 15), LCE = bE1 = aE2, |del| = |E2 - E1(b/a)| ≤ 1 keV, n = 14.51/(Sn/LCE), Sn = 8362.7 keV. In all cases except three, the 576 keV resonance is a partner in the pairs.
References
- TA Brody, J Flore, JB French, PA Mello, A Pandy, et al. (1981) Random matrix physics: Spectrum and strength functions. Rev Mod Phys 53: 385-479.
- ML Mehta (2004) Random Matrices. Academic, New York.
- GE Michell, A Richter, HA Weidenmuller (2010) Random matrices and chaos in nuclear physics: Nuclear Reactions. Rev Mod Phys 82: 2845.
- Sukhoruchkin SI (1972) Deviations from the statistical description of neutron level spacing distributions and stabilizing effect of nuclear shells in positions of nuclear excited states. In: JB Garg, Statistical properties of nuclei. Prenum Press, New York, 215.
- K Ideno, M Ohkubo (1971) Nonrandom Distributions of Neutron Resonance Levels. J Phys Sos Jpn 30: 620-631.
- C Coceva, FC Corvi, P Giacobbe, M Stefanon (1972) J-Dependence of level density in 180Hf and 178Hf from statistical properties of series of resonances with assigned spin. In: JB Garg, Statistical properties of nuclei. Prenum, New York, 447.
- FN Belyaev, SP Bolovlev (1978) Non-statistical effects in the distribution of distances between neutron levels. Sov J Nucl Phys 27: 157.
- S Sukhoruchkin (2000) Tuning Effect in Nuclear Data II. Proc Int Seminar on Interaction of Neutrons on Nuclei, Dubna.
- M Ohkubo (1996) Recurrence of the compound nucleus in neutron resonance reactions. Phys Rev C 53: 1325.
- M Ohkubo (2013) Regular family structures observed in neutron resonance energies: Breathing model of the compound nucleus. Phys Rev C 87: 014608.
- M Ohkubo (2016) Nuclear Poincaré cycle synchronizes with the incident de Broglie wave to predict regularity in neutron resonance energies. Web Conf 122. 13003.
- R Toepke (1974) Messung und Resonanzanalyse von differentiellen elastischen Streuquershnitten an 40Ca. KFK2122 Karlsruhe.
- H. Schopper ( ed) (1998) Low Energy Neutron Physics. Table of Neutron resonance Parameters. Landolt Börnstein new series. Springer.
Author Details
Makio Ohkubo*
N. Resonance Lab, Japan
Corresponding author
Makio Ohkubo, N. Resonance Lab, 1663-39, Senba-cyo, Mito-shi, ibaraki-ken, 310-0851, Japan.
Accepted: September 19, 2019 | Published Online: September 21, 2019
Citation: Ohkubo M (2019) Nuclear Poincaré Cycle Deduced From Regularities in Neutron Resonances in 40Ca and 56Fe - A Comparison with the Semi-Classical Model. Int J At Nucl Phys 4:014.
Copyright: © 2019 Ohkubo M. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Abstract
Thus far, compound nuclei formed by neutron resonance reactions have been believed to exibit quantum chaos form. However, simple integer ratios are widely observed among resonance energies composing families with high accuracy. From these integer ratios, the Poincaré cycles and the nodal resonances are deduced through (Sn/LCE) spectra for 40Ca, and 56Fe, and the neutron resonance energies in the families are described by simple arithmetic expressions based on Sn and several small integers. The breathing model predicts that resonance reactions occur when the Poincaré cycle synchronizes with the de Broglie waves of incident neutrons. A crude model is presented for the compound nucleus composed of several time-quantized normal-modes oscillators with total excitation energy Sn, including the zero-point energy. The Poincaré cycle of the model is compared with the observed values. For 40Ca, and 56Fe, the model Poincaré cycles agree fairly well with the observed cycles.
1. Introduction
Neutron-nucleus interaction cross-sections are indispensable basic data in science and nuclear engineering. These cross-sections have been measured and compiled for almost all possible nuclides using high-resolution neutron time-off light spectrometers with pulsed accelerator neutron sources. In the neutron cross-sections, many fine-structure resonances are observed for all nuclei which are quasi-stable states of a compound nucleus (CN) with excitation energy Ex = Sn + En, where Sn is the neutron separation energy ~ 8 MeV, and En is the neutron kinetic energy in the center of mass system (CMS). Average neutron cross-sections agree well with those predicted by the optical model. The excitation energy is so high that many degrees of freedom will be excited in a CN and mix to create extremely complicated states or chaos. The random matrix theory (RMT) predicts statistical properties of fine-structure resonances: the Wigner (Gaussian orthogonal ensemble) distribution for nearest neighbor spacings of the same Jπ levels, the Porter-Thomas distribution for strengths, and ∆3 statistics for long-range correlations. Many observed resonance data agree well with these predictions. Therefore, the neutron resonance region is long been believed to take a quantum chaos form with no regularity in level spacings or energies [1-3]. However, several studies indicated the presence of many fragments of regular structures buried within neutron resonances with large deviations from RMT distributions [4-9].
Because of the long lifetime of fine-structure resonances, time-periodic quasi-stable-states must exist in a CN. As the model of neutron resonance reactions, we have proposed the breathing model, which states that; the Poincaré cycle synchronizes with the incident neutron de Broglie wave with regular family structures in observed resonances [10,11]. In this study, we propose a new visual method to extract family structures around a nodal resonance Erec which is the Poincaré cycle in this energy region. Observed energy are compaired with of a model CN composed of several oscillators with total energy Sn and the Poincaré recurrence time.
In section 2, we describe the theoretical insight and visual method to obtain the Poincaré cycle and the nodal resonances from (Sn/LCE) spectra. In section 3, we describe the family structures in 40Ca and 56Fe resonances and their models of oscillator ensembles. Discussion and conclusion are presented in section 4.
2. Model of Neutron Resonance Reactions
We have developed the breathing model of CN as a model of neutron resonance reactions. The incident neutron wave is divided into two components. The first component penetrates into the CN and it excites several oscillators to Sn producing the Poincaré cycle of energy Erec. The second component is passing wave or the original de Broglie wave of energy En. When the passing wave synchronizes with the simple harmonics (m/k) (k,m small integers) of the Poincaré cycle, resonance reaction will be enhanced. Time evolution of the compound nucleus at resonance is illustrated in Figure 1.
The CN is assumed to be an ensemble of several normal-modes oscillators whose time period is quantized by τ0 = 2πħ/G, where G is Fermi energy (~ 34.5 MeV). The penetrating component excites the CN to Sn ~ 8 MeV and recurs to the initial condition, causing a flare-up on the CN surface, which is called the doorway, with recurrence time τrec periodically. The recurrence energy of the CN or the energy of the Poincaré cycle is Erec. The following relation is derived [10,11].
Neutron resonances have a tendency to form family (group) structures in which the ratio of the resonance energy En to Erec are in simple integer ratios.
By including the zero-point energy of the Poincaré cycle, better fit of can be obtained.
A method to determine Erec from the observed resonance energies is described below.
For a pair of observed resonances E1 and E2 belonging to the same family of Erec,
E1 = (m1/k1)Erec, and E2 = (m2/k2)Erec. (2.6)
Here, k1, k2 and m1, m2 are small integers. The ratio between E1 and E2, and the least common energy (LCE) of the pair (similar to the least common multiple in arithmetic) are as follows:
E2/E1 = (m2/m1) (k1/k2), (2.7)
LCE = m2 k1 E1 = m1 k2 E2 = m1m2Erec (2.8)
The time period of the Poincaré cycle is elongated for increase of number of oscillators. The ratio Sn/Erec is the time elongation factor of the CN.
Actual procedures to obtain Erec are presented below. Among the ensemble of observed resonances, arbitrary pairs are tested to fulfill the condition E2/E1 = b/a, (a,b = integers ≤ 15) and deviation |del| = |E2-E1(b/a)|. δ is the accuracy of the integer ratios. If |del| < δ, these parameters and (Sn/LCE) are recorded. From the (Sn/LCE) value, Erec can be obtained by the following procedures.
Consider an X-axis starting from 0 to 100, divided into multiple channels of channel width dx = 0.01. For a recorded pair of resonances, one count is added at each channel at integral multiples of x=j × (Sn/LCE), (j=1,2,3....). From Eq.(2.9), integral multiples x = j × (Sn/LCE) meet the Sn/Erec at j = m1m2. For many recorded pairs of resonances, counts are added at positions of different periods (Sn/LCE). However, they coincide at x = (Sn/Erec) and several counts are accumulated, forming large peaks at this point and its integer multiples. In the (Sn/LCE) spectra thus obtained, these peaks visually indicate xp = Sn/Erec, which is the elongation factor of the Poincae cycle. However, as shown below, many peaks appear and adequate selection is needed to obtain true x = (Sn/Erec).
3. (Sn/LCE) Spectra and Nodal Resonances in Light Nuclei
Using high-resolution neutron resonance data of 32S , 35Cl, 37Cl, 40Ar, 40Ca, 54Cr, 56Fe, 64Ni, 68Zn, 90Zr etc., numerical analyses were performed to obtain through (Sn/LCE) spectra by using the software Origin 2018. In the following, results of 40Ca and 56Fe are described. Comparison is made between observed and from the semiclassical breathing model of CN composed of several oscillators.
40Ca + n Sn = 8362.7 keV s-wave resonances
The resonance parameters of 40Ca + n were measured by Toepke up to 2.5 MeV at KFK Karlsruhe [12]. For credibility, analyses were performed on 29 s-wave resonances below En ≤ 1400 keV, whose parameters including observed energies, recoil corrected energies and Γn are listed in Table 1. Among the energies of the 29 resonances, arbitrary pairs with integer ratios E2/E1 = b/a, (a,b < 15) and the deviation |del| = |E2 - E1(b/a)| < 1 keV were selected. Twenty-six cases satisfied these requirements; the values of a, b, E1, E2, del, LCE, Sn/LCE were recorded for these cases. The Sn/LCE values ranged 1.2 ~ 24. On the x-axis of 104 channels of channel widths = 0.01, starting from 0 to 100, one count was added at each channel of periodic positions x = j × (Sn/LCE), where j = 1.2.3... The (Sn/LCE) spectra, obtained from the sum of the counts, showed predominant peaks at Xp = 14.51 and its multiples, as shown in Figure 2. To determine the peaks, averaging was done by multiplying a triangular weight function of half width of 5 channel.
Nine pairs of resonances that consist of the peak at Xp = 14.51 are listed in Table 2; in the table n is the repetition number to attain Xp at a step of (Sn/LCE). In all cases except three, the 576 keV resonance is a partner in the pairs. For these pairs, resonance energies E1, E2 can be expressed in term of 576 keV and a, b, and n, as follows:
E1 = (n/b) × 576 keV, and E2 = (n/a) × 576 keV. (3.1)
As mentioned above, as listed in Table 2, in all pairs except three, a partner is 576keV resonance; that is, 576 keV=(Sn/Xp) = LCE/n. Therefore, we adopt the 576 keV resonance as the nodal resonance, or given by;
As shown in Figure 3, the 576.6 keV resonance is the strongest in the En ≤ 1400 keV region, which is the nodal resonance or the Poincaré cycle in this energy region. When the ratio of incident neutron energy En to is simple integer ratios, reactions are enhanced resonantly. This will be the physical meaning of resonances at 164, 320, 346, 494, 658, 720, and 576 keV.
Furthermore, comparison was made with the breathing model of the CN. Incident neutron excites several time quantized oscillators in the CN with total energy Sn. The Poincaré cycle of these oscillators will synchronize with the incident de Broglie wave. Feasible set of three oscillators, including the 'zero point energy' of the Poincaré cycle, were achieved as follows.
where is the zero-point energy of the Poincaré cycle of the model CN, and G is the Fermi energy treated as a free parameter of 34 ~ 35 MeV. For the set of oscillators (6,20,60), agrees considerably well with the observed value = 576.6 keV (Eq.3.2). Inclusion of the zero-point energy is critical for the numerical consistency. Another oscillator set (6,15) is also feasible with recurrence time 30τ0.
In addition, different families coexist in this energy region. One of the families with a nodal energy = 857 keV comprises 244(2/7), 545(7/11), 979(8/7), and 1286(3/2) keV resonances, where the fractions in parenthesis are the ratios to nodal energy.
Resonances not observed at integer ratios to will be forbidden by several selection rules on the spatial components of wave functions that are not yet known.
The statistical probability was calculated for the appearance of the above-mentioned regular family by assuming random distribution of 29 resonances in the 1400 keV region. The probability that six resonances are on the rational points to the nodal resonance was calculated to be 2.4%. This value is sufficiently small and it is natural to suppose that the family is derived not from randomness but from regularity dominated in this energy region.
56Fe + n Sn = 7646.03 keV s-wave resonances
The resonance parameters of 56Fe + n were measured by Cornelis, et al. and Perey, et al. However, we used evaluated data in the databook provided by Sukhoruchkin [13]. Analyses were performed on 33 s-wave resonances below 830 keV.
Among the energies of the 33 resonances, pairs with integer ratios E2/E1 = b/a, (a,b < 15) and deviation |E2-E1(b/a)| < 0.5 keV were selected. Forty-five cases satisfied these requirements, and the values of a, b, E1, E2, del, LCE, and Sn/LCE were recorded for these cases. (Sn/LCE) spectra showed predominant peaks at Xp = 21.78, as shown in Figure 4.
Seven pairs of resonances compose the peak at Xp = 21.78. In all cases except one, the 351 keV resonance is a partner in the pairs. For these pairs, resonance energies E1, and E2 can be expressed in term of 351 keV, a, b, and n, similar to Eq. (3.1). We adopt the 351 keV resonance as the nodal resonance, that is the Poincaré cycle for resonances in this energy region.
When the ratio of the incident neutron energy En to is a simple integer ratio, reactions are enhanced resonantly. This will be the physical meaning of resonances at 127, 312, 325, 491, 526, 551, and 351 keV. Figure 5 shows resonance strengths gΓn vs. neutron energy.
It is interesting to compare and , selecting several sets of oscillators for the CN model. A feasible set is,
where is the zero-point energy of the Poincaré cycle. Good agreement is attained with the observed value = 351.0 keV (Eq.(3.5)) with only a difference of ~ 4.6 keV.
In addition, in this energy region, different families coexist with the nodal resonance at 491 keV; they are 82(1/6), 184(3/8), 272(5/9), 312(7/11), 350(5/7), 430(7/8), 526(15/14), 536(12/11), 551(9/8), 565(15/13), 653(4/3), and 773(11/7) keV resonances, where the fractions in parenthesis are the ratios to the nodal energy.
4. Discussion and Conclusion
A method is proposed to deduce recurrence energies or the Poincaré cycle, and the nodal resonance of the compound nucleus, using (Sn/LCE) spectra obtained by considering observed neutron resonance energies with simple integer ratios with the nodal resonance energy. For 40Ca and 56Fe, were deduced and the nodal resonances were determined. A crude semi-classical model of neutron resonance reactions of the CN composed of time-quantized oscillators were examined to produce with total excitation energies Sn, including the zero-point energy of the Poincaré cycle. Feasible sets of oscillators provide values that agree fairly well with the observed values . What Eq. (2.4) depict is similar to the harmonic frequencies of resonances in classical physics with non-linearity. However, this model cannot predict resonance strength Γn, Γγ, Jπ, etc., which depend on the spatial components of the wave functions that are not yet known. The Poincaré cycle assumed here must be verified through theoretical and experimental approaches, where the wave functions will be strongly affected by non-linearity in nuclear potentials.
Thus far, under the Random Matrix Theory, only statistical properties of observed resonance data are thought to be meaningful, and much information about individual resonances are disregarded because of lack of strategy to draw physical meaning. Our analyses results indicate that the traditional chaotic view of the neutron resonance region is not accurate and must be replaced with the advanced physics of a regular system. This approach is promising to investigate nuclear dynamic structures through analyses of energy correlations in discrete energy regions.
Further refinements of the breathing model are needed to predict strengths and Jπ. In addition, further high-resolution measurements are needed on the neutron resonance regions.