Autocorrelation of First 4+ Nuclear Energy States

INTRODUCTION

The theoretical description of physical systems often distinguishes between integrable and chaotic systems. Integrable systems, which can be solved exactly using linear equations, contrast with chaotic systems, whose nonlinear dynamics lead to complex, unpredictable behavior.^[1-5] Random matrix theory (RMT) has been proposed to describe quantum systems approaching full classical chaos.^[6]

Nuclear systems often display behaviors that bridge regularity (Poisson-like statistics) and disorder (Wigner-like distributions).^[7-16] Identifying reliable indicators for collective nuclear motion remains a critical research goal. Potential markers include atomic number (Z), neutron number (N), mass number (A), the R_4/2 ratio, P-factor, and deformation parameter (β₂).

Here, we aim to determine the optimal indicator for characterizing nuclear collectivity. Focusing on even-even nuclei for their simplicity and data availability, this work extends prior research on 2⁺ states^[16] to include 4⁺ excitations. Data for the first 4⁺ levels were sourced from the National Nuclear Data Center (www.nndc.bnl.gov), covering 466 even-even nuclei from A = 10 to 256.

Excluding shell-closure influences, lighter nuclei typically show 4⁺ excitations in the (MeV) range, whereas heavier ones drop to hundreds of (keV). Thus, these 4⁺ energies serve as proxies for collectivity trends. We apply autocorrelation analysis to uncover patterns in these datasets.

MATERIAL & METHODS

In this analysis, the sequences of 4⁺ energies, organized by various structural parameters, are treated as discrete time series. The autocorrelation function (ACF) is then applied to measure the internal correlation within each series as a function of the lag, k. For an uncorrelated series, approximately 95% of the ACF values for different k lags are expected to fall within a confidence band of ±2 standard deviations around zero.^[17] The ACF has a value of 1 at k=0 and typically decays toward zero as lags increase. The correlation time, τ_c, is defined as the lag at which the ACF drops to 1/e (approximately 0.37) of its initial value.^[17] A slower decay, indicated by a higher τ_c, is interpreted as a signature of stronger, longer-range correlations associated with collective behavior. The ACF formula is:

where N=466 is the sample size, $E_i$ is the energy of the i-th nucleus, and Ē is the mean energy. To ensure reliability, k is limited to N/4.^[17] Given the broad energy span (keV to MeV) across A=10–256, we analyze the natural logarithm of energies to normalize variability.

RESULTS AND DISCUSSION

The autocorrelation function (ACF) was computed for the natural logarithm of the first 4⁺ energy levels across the 466 even-even nuclei, with the results for each classifying parameter displayed in Figures 1(a-f). Statistical significance, indicated by the 95% confidence level (dotted lines), reveals distinct behaviors. For the Z, N, and A parameters, the ACF decays rapidly, indicating only short-range correlations likely influenced by shell closure effects. A fluctuating, periodic pattern in these ACFs may further reflect the impact of magic numbers. In contrast, classification by the β₂, R_4/2, and P factors produces ACFs with long-range correlations, evidenced by significantly longer correlation times (τ_c).

Close

Figure 1:

ACF versus lag number k for the first 4⁺ states, classified by parameters (a) Z, (b) N, (c) A, (d) β₂, (e) R_4/2, and (f) P. The correlation time τ_c is indicated. The dotted line shows the 95% confidence level. ACF: Autocorrelation function

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Among all parameters, the P-factor yielded the highest ACF value (0.432) and the longest correlation time (τ_c = 74). This result underscores the dominant role of the residual valence proton-neutron interaction in driving collective behavior. Consequently, the P-factor emerges as the most effective indicator for classifying nuclear collectivity, outperforming the other parameters considered in this study.

It is useful to compare the current study with our previous work ^[17] using Table 1 to present the findings succinctly. The aforementioned analysis reveals a consistent pattern of increasing ACF values as we proceed from Z to A, N, β₂, R_4/2, and P, suggesting the generalizability of our findings in selecting the P-factor as a suitable candidate for classifying nuclear collectivity in even-even nuclei.

CONCLUSION

This study employed autocorrelation analysis to investigate collective behavior in even-even nuclei. A dataset of the first 4⁺ energy levels for 466 nuclei was classified according to several nuclear structure parameters: Z, N, A, β₂, R_4/2, and the P-factor. A logarithmic transformation was applied to the energy values to stabilize variance across the series before calculating the ACF with a lag number of k = N/4.

The results demonstrate that the P-factor yields the largest ACF enhancement and the longest correlation time (τ_c), underscoring its superiority for identifying collective trends. This finding confirms that the P-factor is an effective parameter for grouping nuclei with similar collective properties. Furthermore, it successfully generalizes the conclusions of our previous work on 2⁺ states to the higher 4⁺ energy state, reinforcing the robustness of this approach.