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The Generalized Liquid Drop Model Alpha-Decay Formula: Predictability Analysis and Super-Heavy Element Alpha Half-Lives

The predictive accuracy of the generalized liquid drop model (GLDM) formula for alpha decay half-lives has been investigated in a detailed manner and a variant of the formula with improved coefficients is proposed. The method employs the experimental alpha half-lives of the well-known alpha standards (REFERENCE) to obtain the coefficients of the analytical formula using the experimental Qalpha values (the DSR-E formula), as well as the finite range droplet model (FRDM) derived Qalpha values (the FRDMFRDM formula). The predictive accuracy of these formulae were checked against the experimental alpha half-lives of an independent set of nuclei (TEST) that span approximately the same Z,A region as the standards and possess reliable alpha spectroscopic data, and were found to yield good results for the DSR-E formula but not for the FRDM-FRDM formula. The two formulae were used to obtain the alpha half-lives of super-heavy (SHE) and heavy nuclides where the relative accuracy was found to markedly improve for the FRDM-FRDM, which corroborates the appropriateness of the FRDM masses and the GLDM prescription for high Z,A nuclides. Further improvement resulted, especially for the FRDM-FRDM formula, after a simple linear optimization over the calculated and experimental half-lives of TEST was used to re-calculate the half-lives of the SHE and heavy nuclides. The advantage of this optimization was that it required no recalculation of the coefficients of the basic DSR-E or FRDM-FRDM formulae. The halflives for 324 medium-mass to super-heavy alpha decaying nuclides, calculated using these formulae and the comparison with experimental half-lives, are presented.


Introduction
A number of recent advances in the synthesis of very heavy and superheavy elements (SHE) [1,2] as well as highly neutron deficient medium mass nuclei [3], have spurred theoretical investigations into the decay modes and masses of these nuclei. These investigations aim to test and advance ideas on the extant nuclear structure for heavy and high Coulomb repulsive force containing nuclei. From the stand-point of heavy element chemistry, the isolation and the study of the properties of exotic heavy elements and their compounds serve to provide unique insights into the structure, bonding and reaction energies of massive multiply charged atomic and molecular species [4] Often the main and sometimes the only decay mode of these nuclei is alpha decay.
For the SHE, the product nuclides of hot-fusion heavy ion reactions closest to the shell model predicted Island of Stability, decay by the emission of alpha particles. Chains of successive alpha decays are terminated by spontaneous fission as the shell-stabilized region is left behind [5,2].Thus identifying and characterizing the alpha decay sequences form a crucial part of the identification of SHE. Theoretically the mechanism is described by quantum mechanical tunneling through the potential energy barrier leading from the mother nucleus to the daughter nucleus and alpha particle. Consequently the predicted half-lives remain very sensitive to the shape and energetics of the barrier which as such, serve to test the particular theoretical model of the potential energy surfaces of these exotic nuclei. Various theoretical prescriptions since 1930 [6] have been proposed. Some of these as well as some empirical observations have been reduced to analytical formulae [7][8][9][10][11][12] that connect Q α , and the Z,A of the parent nuclide and wherein coefficients are typically obtained from fits to known alpha half-lives. The underlying model's description of the potential barrier lies implicit in the formula's functional form and its coefficients and this in part determines the merit of the formula.
For the experimentalist, the analytical formula is especially useful because it permits easy incorporation into experimental data processing/analysis systems as well as the possibility of the upgrade of the coefficients of the formula as the experimental data base expands and/or gets refined. Ken Moody representing the Dubna-Livermore collaboration on SHE synthesis [2], mentions their use of the Geiger-Nutall relationship [7] to check their assignment of the Z of the composite system to the parent nuclides of each alpha decay chain. The experimental investigation of these exotic alpha emitters is constrained by the smallness of the number of atoms formed, the shortness of half-lives as well as the varying degrees of uncertainty in the theoretical prescriptions and formulae [13].
A systematic analysis of the predictive accuracy of several well-known or recent analytical formulae has to date received relatively insufficient attention. Often the degree of closeness of match between the predicted half-lives and the experimental half-lives of newly discovered exotic alpha emitters is taken to be a measure of the goodness of the formula towards the prediction of unknown half-lives [14,15]. Such an accuracy test is compromised because the experimental data of the exotics are subject to substantial uncertainties, because the coefficients of the formula have been arrived at by a fitting procedure that involves the experimental data of a large number of alpha emitters many of whose mass and spectroscopic data are not sufficiently well-known and because for many if not most of the exotics, the Q α have to be taken from systematics or theoretical tabulations which have their inherent errors for the particular mass range under consideration [18]. This study is an attempt to investigate in a systematic and detailed manner the reliability of the recent alpha decay analytical formula of Royer et al [14] in terms of its predictive accuracy. Concomitantly, it devises a method for the improvement of the reliability of the formula's coefficients and explores a simple optimization procedure for the application of the resulting formula to the half-lives of heavy and SHE alpha emitters.

The Generalized Liquid Drop Model Formula for Alpha Decay Half-Lives
Recently Royer et al [14][15][16][17] have described the potential barrier in alpha decay in terms of a quasi-molecular shape path within a Generalized Liquid Drop Model (GLDM) including the proximity effects between nucleons in a neck, and the shell effects given by the Droplet Model. The difference between the experimental Q α and the GLDM calculated one was empirically corrected by adding it to the macroscopic potential energy of the parent nucleus with a linear attenuation factor that vanished at the contact point of the nascent fragments. The half-lives were deduced from the WKB barrier penetration probability as for spontaneous asymmetric fission. The theoretically calculated logarithms of half-lives for known alpha emitters wherein the experimental Q α were used, agree well with the logarithms of the experimental half-lives, with a total root mean square deviation (RMSD) of 0.63, which was taken as evidence of the essential soundness of the model. However, for the subset for the heavy nuclides (Z≥ 100) whose Q α are estimated from systematics, the agreement is comparatively worse with an RMSD of 1. 16   In this work we are concerned only with the reduction of Royer's theoretical calculations to the analytical formula, above. While the large number of 373 nuclides is beneficial for the statistics of the fitting procedure, it is not possible to get a clear idea of the closeness of match between the formula calculated half-lives and the experimental ones because any reasonably large set of well-studied alpha emitters will have members already included in the set of 373 nuclides and will therefore not be independent.
Moreover for a sizable number of these 373 emitters particularly those at the extrema of the (Z,A) scale, the alpha decay properties possess significant uncertainties and for the heaviest nuclides the Q α values are those reported only from systematics. These uncertainties will affect the values of the constants evaluated for the analytical formula.
Slight variations in the coefficients in eqn. 1 as also the Q α , will have a relatively large effect on the calculated half-lives because of the logarithmic relationship. Most of these nuclides with uncertain nuclear data in [14], have undetermined or poorly determined alpha branching fractions (e.g. 110,111 Xe, 111 I and 189 Bi) while most of the heavy nuclides with Z > 100 have Q α surmised by systematics.

Construction of the Method for the Predictability Test
We use the term "predictability" to connote the extent of accuracy of the GLDM formula in its prediction of the half-lives of an independent set of nuclei, i.e. nuclei whose data have not been used to arrive at the values of the coefficients a, b and c. This test of accuracy necessitates the experimental nuclear data of this independent set of nuclides, the 'TEST data' set, to be also well known. The purpose of this work is to test the predictability of the GLDM formula over a defined Z,A range using such a method with the coefficients of the formula having been arrived at using accurate and highly reliable data of a set of "REFERENCE" nuclei that cover approximately the same Z,A range.
Towards this objective we first establish a set of standard nuclei whose alpha decay properties are very well known. These, called REFERENCE nuclides, are taken to be the set of nuclides recommended as alpha energy and intensity standards for the calibration of alpha particle measurements [18a]. The experimental Q α (MeV), total halflife in seconds (T) and the percentage alpha branching fraction "a", along with their respective errors were obtained from the  18] to be those that also possess well-known spectroscopic data, the ad hoc indicator of which was taken to be the criterion that the nuclide possess at least 5 references on the determination of its spectroscopic data.
The coefficients a, b and c of eqn 1 were arrived at after a multi-variable regression fit [19] using the T(E), Q α and (Z,A) of the REFERENCE nuclides. The magnitude of statistical precision error is expected to be higher than those calculated from a much larger data set even one with some inaccurate elements. This is an inevitable constraint as it is not possible to get an extremely large highly reliable set of data.
However, this work is not concerned with the establishment of a more precise formula but rather one whose underlying data confer a high degree of confidence and which thereby can serve as "calibration formula" for the set of TEST nuclides. respectively. For the purpose of distinguishability, the GLDM formula (eqn 1) using these coefficients is termed the DSR formula whereas the GLDM formula with coefficients as derived in [14] (cf section 2.1) is termed the R formula.
The TEST data set, were taken to be nuclides that covered a (Z,A) range close to the REFERENCE nuclides to reduce the possibility of the influence of Z,A dependent differences in the underlying nuclear parameters in the data analysis. These nuclides, labeled as TEST, while not alpha particle measurement standards, were nonetheless chosen to be those with well-known experimental data [18]. The criterion of selection was that the nuclide possess at least 3 references in [18] for the experimental determination of its spectroscopic data. The half-lives for the TEST nuclides, T(FRDM-FRDM) were calculated from eqn 1 using these coefficients and the FRDM Q α . Additionally, the TEST nuclide half-lives were also calculated using the standard DSR and R coefficients and the FRDM Q α , labeled as T(DSR-FRDM) and T(R-FRDM).
As measure of accuracy we take the square of the relative error or fractional deviation, FDsq, defined as, Where, "Calc" stands for the mode of calculation used to arrive at the half-life: T for theory derived, DSR and R for calculations using the DSR and R analytical formulae respectively and FRDM for the GLDM formula with coefficients derived using FRDM Q 3 Results and Discussion in the set of 373 nuclides from which coefficients of the R formula had been derived in [14].

Analysis of Predictability of the GLDM Formula
The root mean squared relative accuracy of the DSR formula using experimental The analytical formula finds particular application in the predictions of half-lives of exotic nuclei whose Q α are usually not reliably known. In the following we address the question of the quality of predictability of the GLDM analytical formula when theoretical masses have to be used. Royer [14] uses the Thomas-Fermi model derived Q α values in the R formula to obtain the half-lives of SHE. For some SHE the calculated half-lives were within the range of uncertainties, albeit quite large, of the experimental data extant at that time while they were off by several orders of magnitude for the e-o SHE. Table 2 shows the FRDM Q α and the half-lives of the TEST nuclides obtained using the DSR

The GLDM Formula and the Half-Lives of SHE
The DSR-E formula's coefficients have been derived from reliable data (the REFERENCE nuclides) and therefore are 'constant' in the sense that there will be little likelihood of the need for occasional data-base dependent upgrades. On the other hand, the R formula's coefficients have been obtained from a less reliable but wider Z,A spanning database which results in a smoothing over of the effects due to the variation of the nuclear parameters across this Z,A range. The relative predictive accuracies of the 2 prescriptions are examined via the calculated half-lives for the heavy nuclides with Z ≥99 and recently discovered SHE [2]. Table 3 lists the experimental data and half-lives calculated by the DSR-E, R, FRDM-FRDM as well as the optimized DSR-E and FRDM-FRDM expressions discussed below. The data for the e-e and e-o SHE with Z ≥ 110 were taken from [2], the heavy nuclides from [18] and Uuu( 272 111) from [14]. The errors on the T(E) values could not be ascertained as for many nuclides in the data base, the full spectroscopic data are not known. Also for many, the Q α have been reported only from systematics. Thus the root mean fractional deviation squared (RMFDsq) values serve as approximate guides only to the relative accuracy.

Conclusion
The accuracy of the generalized liquid drop model, GLDM, of Royer et al [14][15][16][17] towards the prediction of alpha half-lives, has been tested in a detailed manner. The analytical formula of [14] was modified resulting in the DSR-E formula that has the advantage of possessing coefficients derived from a reliable data base (REFERENCE).
Using this formula, the predictive accuracy of the GLDM formula was checked against the experimental half-lives of a set of nuclides whose alpha decay data are well-known (the TEST) and was found to be satisfactory. The coefficients of the formula based on theoretical Q α , the FRDM-FRDM formula, were also obtained using the REFERENCE nuclides, where FRDM is the finite range droplet model formalism of nuclear masses [20]. Its predictive accuracy, checked in a similar manner, was found to be unsatisfactory. The results of this work suggest that (1) the ansatz of selecting 2 data sets with well-known spectroscopic data, one for calibration and the other for test, is a valid method for the check of accuracy of a theoretical prescription for the decay half-life (2) the GLDM analytical formulae with coefficients derived from a limited but highly accurate data set, result in good agreements between model predicted and experimental alpha decay half-lives; (3) that the use of model derived Q α necessitates derivation of the coefficients of the alpha decay formula using the same model for the Q α for reasons of consistency; (4) that the process of simple linear optimization of the two variants of the GLDM formula using experimental or model derived Q α , result in the best predictive accuracies for heavy nuclides and SHE and finally (5) that the GLDM formula with FRDM masses produce better predictive accuracy for the highest Z,A alpha emitters.
Furthermore, this method of calibrating the calculation against a 'standard' data set, allows the comparison between two independent (Z,A) regions as well as a simple linear optimization procedure whereby the predictive accuracy of the formula can be improved without altering the coefficients of the calibrated formula. Effectively this provides a possible means of ameliorating the statistical limitation of the narrower data set of the 'standard' without changing its characteristics. While this work has centred on the GLDM formula, the same methodology could be applied to other analytical formulae with coefficients derived from fits to experimental values.  [18] and a few others that also possess well-known alpha spectroscopic data. The four nucleon parity sets, e-e, e-o, o-e and o-o are labeled at the top each sub-section of the table. The experimental values were derived from [18] and the theoretical alpha half-life calculated from the Generalized Liquid Drop Model (GLDM), was taken from [14]. All Q values are in MeV and all half-lives are in seconds. The explanation of the entries in the table are given below.

Parent (column 1):
The parent alpha decaying nuclide Z,A (column 2): The charge and mass numbers of the parent nucleus.
The experimental Q value of alpha decay [18].
The Q value of alpha decay calculated using the Finite Range Droplet Model (FRDM) tabulation [20] T(E) (column 5): The experimental partial half-life of alpha decay.

SE (T(E)) (column 6):
The standard error on the experimental half-life of alpha decay. The values of T(E) and the propagated error SE(T(E)) were calculated from the experimental total halflife and α branching fraction and their standard deviations [18].

T(T-E) (column 7):
The GLDM theory calculated α half-life [14].   [18] and the calculated alpha half-lives were arrived at using the analytical GLDM formula [14], with coefficients as derived in this work in which both experimental and FRDM [20] Q α values were employed. See the explanation of Table 1 for the expansion of the abbreviations. All Q values are in MeV and all half-lives are in seconds. The explanation of the entries in the table are given below.

Column 1 to Column 7:
Entries are the same as in Table 1 T(DSR-E) (column 8): The DSR-E calculated α half-life. This variant of the GLDM analytical formula, possesses coefficients that have been derived using the Q α (exp) of REFERENCE.. The calculated α half-life, T(DSR-E), of the unknown nuclide is obtained by using its Q α (exp) in the resulting analytical formula.   [2], b) from [14] and the rest as before from [18]. The calculated α half-lives were arrived at using the analytical GLDM formula [14] with coefficients as derived in this work in which both experimental and FRDM [20] Q α values were employed as well as the expressions resulting from the linear optimization of the calculated half-lives of TEST to its experimental values. See the explanation of Table  1 for the expansion of the abbreviations. All Q values are in MeV and all half-lives are in seconds. The explanation of the entries in the table, are given below.

Column 1 to Column 5:
Entries are the same as in Table 1 T(R-E ) (column 6): The R-E calculated α half-life. This is Royer's [14] analytical formula derived from the GLDM.    Table. Even                 Table. Even     Table. Even    Table. Even    Table. Even Atomic and mass numbers, the experimental and calculated Q value of alpha decay, the experimental and calculated values of the alpha decay half-lives, for the 22 heavy and super-heavy elements (SHE).
See p. 25 for Explanation of Table. Odd