Modified empirical formulas and machine learning for $\alpha$-decay systematics

The journey of $\alpha$-decay, starting from Rutherford followed by Geiger-Nuttall law [1] in 1911 along with quantum mechanical formulation by Gamow [2] and, Guerney and Condon [3] in 1928 to till this year 2020 [4, 5, 6, 7, 8] dominates the investigations of superheavy nuclei in such a way that most of the experimental and theoretical studies of superheavy nuclei are concentric primarily on the $\alpha$-decay mode. Study of $\alpha$-decay chains plays a crucial role for synthesis of new elements in the laboratory and also to investigate the nuclear structure properties [9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. The synthesis of superheavy elements in laboratories at GSI [9, 19], RIKEN [20] and JINR [10, 21, 22] is mainly governed by cold-fusion reactions between ²⁰⁸Pb, ²⁰⁹Bi and beams of A$>$50 [10] or hot-fusion reactions between long-lived actinide nuclei from ²³⁸U to ²⁴⁹Cf and ⁴⁸Ca beams [11]. As an extension of the periodic chart, decay chains of ²⁹³117 and ²⁹⁴117 have been reported by fusion reactions between ⁴⁸Ca and ²⁴⁹Bk with 11 new nuclei [21] and also of ²⁹⁴118 by ²⁴⁹Cf+⁴⁸Ca reaction [23]. In addition, afterward, eleven new heaviest isotopes of elements Z$=$105 to Z$=$117 are identified validating the concept of the island of enhanced stability for superheavy nuclei (SHN) [24]. For the synthesis of elements with Z$\geq$118 the attempts and progress are still going on and in near future few more SHN are expected to be detected [15, 25].

In recent times, experimental works are performed on the Dubna gas-filled magnetic recoil separator (GFS) to study the reactions of the formation of ²⁹⁴Ts and ²⁹⁴Og nuclides along with future possibilities of synthesizing new elements with Z$=$119 and Z$=$120 [8]. Similarly, the observation of a new decay chain of ²⁹⁴Og along with prospects for reaching new isotopes ^295,296Og are reported [26]. Many properties of the decay of the heaviest nuclei with Z$=$112–118 indicate a considerable stabilizing effect upon approaching the island of stability N$\sim$184 [11]. However, up to now the heaviest experimentally known nuclei ²⁹⁴Ts and ²⁹⁴Og have neutron number N$=$177 and N$=$176, respectively. Therefore, future experiments are expected to synthesize heavier elements which may have the neutron number between N$\sim$170 and N$\sim$184. With this in view, the study of decay chains of Z$=$118 and 119 with neutron number 168$\leq$N$\leq$184 may be of extreme importance and crucial for experimental point of view, which is precisely the aim of this article.

The study of $\alpha$-decay chains consists of the determination of $\alpha$-decay half-life ($log_{10}T_{1/2}$) which is first formulated as linear dependence on reciprocal of the square root of $\alpha$-decay energy ($\sqrt{Q}$) [1]. In 1966, Viola and Seaborg [27] predicted a simple formula that is based on the Gamow model by including the intercept parameters of linear dependence on the charge number of the daughter nucleus and also modified in Refs. [28, 29]. Afterwards, Brown [30], Royer [31] and Ren et al. [32] have formulated the empirical formulas by studying the experimental variation of the logarithmic half-lives. Later on, Ni et al. [33] have derived the empirical formula (NRDX formula) for alpha decay half-lives from the WKB barrier penetration probability with certain approximations. Likewise, Qi et al. [34] proposed a linear universal decay law (UDL) from the microscopic mechanism of the charged-particle emission. In addition to the formulas mentioned above, some semi-empirical relationships including microscopic effects, such as the universal (UNIV) curve, and a semi-empirical formula based on fission theory (SemFIS), were proposed by Poenaru et al. which are subsequently modified in Refs. [35] and [36] proposing new names NUNIV and SemFIS2, respectively. Recently, many of these formulas have been modified by adding asymmetry terms and referred as modified Royer formula (Akre formula) [37], modified UDL formula (MUDL) [38], modified scaling law of Brown (MSLB) [39], modified Viola-Seaborg formula (MVS) [39], and new Ren A formula [40]. Few of the above mentioned formulas are refitted using latest experimental alpha decay half-lives viz. (i) modified Royer GLDM (Rm), modified Viola-Seaborg (VSm1 and VSm2) and modified Sobiczewski-Parkhomenko (SPm1 and SPm2) formulas [41], (ii) modified version of the Brown empirical formula (mB1 and mB2) [42] (iii) new universal decay law (NUDL), new Royer (NRo) and new Sobiczewski-Parkhomenko (NSP) formulas [43]. Moreover, Manjunatha et al. have presented a formula [44] based on second-order polynomial fitting of $Z^{0.4}_{d}/\sqrt{Q}$ which is subsequently modified in one of our recent work [45].

Besides the above-mentioned formulas, in 2004, Horoi et al. [46] have proposed the scaling law for cluster decay and in 2005, Parkhomenko and Sobiczewski [29] have included excitation energy for odd-A and odd-odd nuclei in an empirical formula for Z$>$82. Both these formulas may be modified further and the parameters may be adjusted with the fitting of updated experimental data which is one of the motives of the present work.

Another decay mode that competes with $\alpha$-decay and acts as a decider to terminate the $\alpha$-decay chains is spontaneous fission (SF). The first semi-empirical formula for SF was proposed by Swiatecki [47] in 1955 and afterward in 2005, Ren et al. [48] have generalized formula for spontaneous fission half-lives of even-even nuclei in their ground state to both the case of odd nuclei and the case of fission isomers. Later on, Xu et al. [49] have proposed empirical formula of SF for even-even nuclei which is followed by the semi-empirical formula given by Santosh et al. [50]. In 2012, Karpov et al. [51] have proposed a new formula based on barrier height which is modified with even-odd corrections by Silisteanu et al. [52]. In 2015, a modified formula is proposed by Bao et al. [53] for determining the spontaneous fission half-lives based on Swiatecki’s formula, including the microscopic shell correction from FRDM-1995 [54] and isospin effect. This formula was able to reproduce the experimental data quite well, however, it may be further modified with shell plus pairing correction from FRDM-2012 [55] which is also an object of present work.

In recent times, machine learning (ML) algorithms [56, 57] have participated as an alternate and powerful tool to study and predict a complex data of physics [58, 59, 60]. In particular, machine learning methods are employed in nuclear physics for prediction of nuclear masses [61, 62], binding energies [63], $\beta$-decay [64] and charge radii [65]. Since the predictions in the superheavy realm by various theories and empirical/semi-empirical formulas are still not so precise, therefore, machine learning methods could be very useful to predict decay modes, decay chains, and half-lives of unknown territory of superheavy nuclei. This possibility has invoked us to apply machine learning methods for the prediction of $\alpha$-decay, SF and for the determination of half-lives of known and unknown nuclei to visualize the merit of this alternate approach in superheavy region.

The paper is organized as follows. In section II we provide the original formulas and their modified versions with concerned coefficients for $\alpha$-decay and spontaneous fission (SF) half-lives. These formulas are fitted using $scipy.optimize.curve\_fit$ library [66] along with the Levenberg-Marquardt algorithm which is used for non-linear least-squares fit [67]. In Section III we give a brief outline of the all machine learning algorithms employed. In section IV we first compare results of modified formulas with already established formulas and then compare our results of half-lives for a few experimentally known decay chains. Then, we predict probable decay modes for the nuclei in the range 112$\leq$Z$\leq$118 which is followed by a prediction of potential decay chains of ^286-302Og and ^287-303119.

In the beginning, to test the predictive power of few known and mostly used $\alpha$-decay empirical/semi-empirical formulas, we have calculated $\alpha$-decay half-lives using experimental and evaluated Q_α [69] for 366 nuclei within the range 82$\leq$Z$\leq$118. The formulas used are Sobiczewski [29], NRDX [33], new universal curve (NUNIV) [35], semi-empirical formula based on fission theory (SemFIS2) [36], Akre formula given by Akrawy et al. [37], modified UDL formula [38], MSLB formula [39], modified Viola-Seaborg formula [39], Manjunatha present formula [44], Horoi [46], new Ren A formula [40], modified Royer GLDM (Rm) [41], modified Viola-Seaborg (VSm1 and VSm2) [41], modified Sobiczewski-Parkhomenko (SPm1 and SPm2) formulas [41], modified version of the Brown empirical formula (mB1 and mB2) [42], new universal decay law (NUDL), new Royer (NRo) and new Sobiczewski-Parkhomenko (NSP) formula [43]. In addition, the modified formulas in the present work i.e. modified Horoi formula (MHF-2020) (set-1 and set-2) and modified Sobiczewski formula (MSF-2020) are also tested for the same set of nuclei.

The root mean square error (RMSE), mean deviation ($\overline{\delta}$) and the index F$=$10${}^{\overline{\delta}}$ expressed by the following equations are deduced for all the above mentioned formulas and mentioned in Table 4. The total number of fitted coefficients are also listed in front of each formulas. A justified comparison among formulas with coefficients $>$10 leads the modified formulas with a good merit. Also, the RMSE and $\overline{\delta}$ are plotted in Fig. 3 to depict a comparison of the above mentioned formulas. It is worth to mention here that few formulas with comparatively lesser number of coefficients (Sobiczewski, NUDL etc.) still possess a good accuracy for the calculation of half lives.

It is gratifying to note from Table 4 and Fig. 3 that the modified formulas introduced in present work, result with minimum error and deviation comparative to other 21 considered formulas. In fact, comparison shown in the Table 4 for all the statistical parameters certifies the applicability of the MHF-2020 (set-1 and set-2) and MSF-2020, and therefore will be employed for prediction of decay modes, to calculate $\alpha$-decay half-lives and to construct decay chains in the unknown territory of superheavy land.

In a similar manner, to find more precise formula for spontaneous fission: the mode which compete with $\alpha$-decay, we apply the formula given by Ren et al. [48], Xu et al. [49], Santosh et al. [50], Karpov et al. [51], Silisteanu et al. [52], Bao et al. [53] along with the modified Bao formula of present work. Spontaneous fission half-lives are calculated by using above all formulas for 49 nuclei for which SF-half-lives are known and consequently a few statistical parameters are obtained which are listed in Table 5 and also shown in Fig. 4. The comparison displays improvement in the prediction of SF half-lives after the modification in the formula of Bao et al. [53] which earmark the use of modified Bao formula (MBF-2020) to analyze the competition between $\alpha$-decay and SF.

As mentioned above, four methods of machine learning i.e. Decision Trees (DTs), Random Forest (RF), Multilayer Perceptron (MLP) neural network, and XGBoost are tested for $\alpha$-decay and SF half-lives considering 366 and 49 nuclei, respectively. We apply 5-fold cross-validation, and accordingly, the dataset is split into 5 groups. Using this 5-fold cross-validation, root mean square error (RMSE) and mean deviation ($\overline{\delta}$) of considered methods of machine learning are mentioned in Table 6.

The Table 6 establishes MLP neural network and XGBoost methods as the more appropriate method of machine learning among the considered ones because for these two methods the RMSE and $\overline{\delta}$ for $\alpha$-decay and SF half-lives are found very similar to what are found for modified empirical formulas (please see Tables 4 and 5). Therefore, in the following, we will employ MLP and XGBoost for the prediction of $\alpha$-decay half-lives. For SF half-lives, since the training data is very limited therefore predictions of MLP neural network for such a small dataset are not found reasonable. As a result, we opt the Random Forest (RF) method to predict SF half-lives along with XGBoost algorithm.

With the above-mentioned selection of empirical formulas of $\alpha$-decay and SF half-lives along with machine learning methods, in Table 7, we predict the decay modes of nuclei in the range 112$\leq$Z$\leq$118 to demonstrate the predictive power of modified formulas and machine learning methods. It is important to mention here that in all the formulas of $\alpha$-decay considered here (3 Proposed$+$21 others), the contribution of centrifugal potential is not included. The calculations show that favored $\alpha$-decay in which the orbital angular momentum taken by the $\alpha$-particle is zero ($l$$=$0), can be efficaciously described by the considered formulas. However, for unfavored $\alpha$-decay ($l$$\neq$0) one needs to take into account the $l$ dependence which is left for our upcoming work.

In Table 7, a few nuclei from 112$\leq$Z$\leq$118 are taken for which change in angular momentum $l$$=$0 (favoured) and the values of Q_α are evaluated from latest database [69]. Few available $log_{10}$T_1/2 and decay modes are also tabulated for comparison. To predict $\alpha$-decay half-lives we use modified Horoi formula (MHF-2020) (set-1 and set-2) and modified Sobiczewski formula (MSF-2020) which are constructed in this present work. Also, to predict SF half-lives we employ modified Bao formula (MBF-2020) reported in this work. These formulas are utilized to analyze the competition of $\alpha$-decay with SF and consequently to predict probable decay mode. The same kind of predictions are also described in Table 7 by using machine learning methods as an alternate approach.

Table 7 itself expresses the predictive power of modified formulas and also the use of machine learning methods as the decay modes are anticipated effectively. In fact, not only the decay modes are found in an excellent match with available experimental/estimated decay modes but also the half-lives are found in agreement with experimental half-lives. In addition, we have compared our estimation of half-lives for the experimentally known decay chains [21, 23, 76, 77]: ²⁹⁴Og, ²⁹⁴Ts, ²⁹³Ts, ²⁹¹Lv, ²⁸⁸Mc and ²⁸⁷Mc in Fig. 5. The $\alpha$-decay half-lives are calculated by employing MHF-2020 (set-1 and set-2) and MSF-2020 which are constructed in this present work along with machine learning methods: XGBoost and MLP. Satisfyingly, these half-lives are found in a reasonable match with the experimental decay half-lives mentioned in Fig. 5 with error bars.

This agreement countenances the use of formulas and methods reported in the present work to estimate decay modes, decay chains, and half-lives for the unknown territory of superheavy nuclei to accompaniment the crucial experiments eyeing on heavier elements with Z$\geq$118 [15, 25, 8, 26]. With this in view, we employ the above-mentioned formulas and machine learning methods to construct the possible decay chains for isotopes of Og and Z$=$119 which are of current interest. The decay chains are constructed for neutron number lying between 168$\leq$N$\leq$184 as observed from experimental decay chains [21, 23, 76, 77] and speculated for island of stability at N$\sim$184 [15, 25, 8, 26]. To apply empirical formulas for calculations of $\alpha$-decay half-lives the experimental and evaulated Q_α [69] are used. For the rest of the nuclei, the values of Q_α can be taken from theoretical approaches. In this regard, we have performed calculations using relativistic mean-field theory [78, 79, 80, 81] for the nuclei in the range of 111$\leq$Z$\leq$118. The values of Q_α are also taken directly from FRDM [82], RCHB [83] and KTUY [84] mass tables for comparison. For determination of Q_α, one can also use the very efficient five-parameter formula derived from LDM considerations by Dong et al. [85, 86], which is given by:

In addition to the above treatments for Q_α, we have applied machine learning methods as an alternate path. To train a machine, we have used all experimental/evaulated Q_α values ($\approx$709) between 82$\leq$Z$\leq$110 with the 5-fold cross-validation as described above. Once again XGBoost is found with minimum RMSE among the considered machine learning methods and, therefore, we have predicted Q_α values for the nuclei in the range 111$\leq$Z$\leq$118 using XGBoost algorithm. The errors in the values of Q_α for the nuclei in the range 111$\leq$Z$\leq$118 calculated by RMF [78, 79, 80, 81], FRDM [82], RCHB [83], KTUY [84], Dong et al. formula [85, 86] and XGBoost are plotted in Fig. 6. RMSE for all these approaches are also mentioned in the corresponding panels.

It is clear from Fig. 6 and mentioned RMSE values that XGBoost is found closer with experimental/evaulated Q_α values comparative to the theoretical approaches and the formula. Hence, for the prediction of half-lives for Og isotopes and Z$=$119 isotopes, we will opt Q_α values from XGBoost algorithm where experimental Q_α are not available. In this way, with the more accurate Q_α, we expect a more reliable prediction of $\alpha$-decay half-lives using more precise formulas (MHF-2020 & MSF-2020, please see Table 4) reported in the present work. In Tables 8 & 9, we show decay chains of Og isotopes in the range 286$\leq$A$\leq$302. As mentioned above, to speculate chances of favoured and unfavoured $\alpha$-decay, we have also mentioned change in the angular momentum ($l$) calculated from selection rule [87, 88] based on parity and spin of parent and daughter nuclei taken from latest evaluated nuclear properties table NUBASE2016 [89] or P. Möller [90]. More detailed analysis on favoured and unfavoured $\alpha$-decay will be covered in our upcoming work.

Asterisk (*) values of Q_α are taken from the XGBoost algorithm whereas all other values are experimental/evaluated [69]. Using these Q_α values, $\alpha$-decay half-lives ($log_{10}$T${}_{1/2}^{\alpha}$(s)) are deduced by modified Horoi formula (MHF-2020) (set-1 and set-2) and modified Sobiczewski formula (MSF-2020), and also with MLP and XGBoost algorithm of machine learning. Spontaneous fission half-lives ($log_{10}$T${}_{1/2}^{SF}$(s)) are also mentioned which are calculated by using modified Bao formula (MBF-2020). Consequently, decay modes are predicted for all the chains which are compared with available experimental/estimated decay modes [69]. Similar calculations and prediction are tabulated for Z$=$119 isotopes in the range 287$\leq$A$\leq$303 in Tables 10 & 11.

From Tables 8 to 11, it is indulging to note that our prediction of decay modes and half-lives are in an excellent match with available experimental data. ^286-291Og and ^{287-296,302,303}119 are found with very long $\alpha$-decay chains (6$\alpha$/7$\alpha$) and are found as potential candidates for future experiments. As far as the current experimental facilities are concerned, ⁵⁰Ti+²⁴⁹Bk and ⁵⁰Ti+^249-251Cf reactions could lead to the discovery of ^295,296119 and ²⁹¹Og [8] which are found with 7$\alpha$ decay chains as per our predictions. For ⁴⁸Ca beam, the available heaviest target material is ²⁵¹Cf (N = 153) which could lead to the production of ²⁹⁵Og, ²⁹⁶Og, and ²⁹⁷Og [22] and from our prediction, these isotopes of Og are indeed found with 3$\alpha$/4$\alpha$ chain and their half-lives are found within the range of current experimental facilities. Therefore, our predictions from the modified formulas are expected to provide more extensive and precise information for the upcoming experiments to reach the boundaries of the island of stability.