Alpha decay calculations with a new formula

TL;DR

This paper introduces a new semi-empirical formula for alpha decay half-life calculations, derived from Royer's relationship, and compares its performance with established models across multiple datasets.

Contribution

The paper presents a novel semi-empirical formula for alpha decay half-lives, fixed by experimental data, and demonstrates its competitive accuracy against well-known models.

Findings

The new formula performs well across different datasets.

It competes with established models like semFIS and UNIV.

It is simpler yet effective in predicting alpha decay half-lives.

Abstract

A new semi-empirical formula for calculations of $\alpha$~decay halflives is presented. It was derived from Royer relationship by introducing new parameters which are fixed by fit to a set of experimental data. We are using three sets: set A with 130 e-e (even-even), 119 e-o (even-odd), 109 o-e, and 96 o-o, set B with 188 e-e, 147 e-o, 131 o-e, and 114 o-o, and set C with 136 e-e, 84 e-o, 76 o-e, and 48 o-o alpha emitters. A comparison of results obtained with the new formula and the following well known relationships: semFIS (semiempirical based on fission theory), ASAF (analytical superAsymmetric fission) model, and UNIV (universal formula) is made in terms of rms standard deviation. We also introduced a weighted mean value of this quantity, allowing to compare the global properties of a given model. For the set B the order of the four models is the following: semFIS, UNIV, newF, and ASAF. Nevertheless for even-even alpha emitters UNIV gives the 2nd best result after semFIS, and for odd-even parents the 2nd is newF. Despite its simplicity in comparison with semFIS the new formula, presented in this article, behaves quite well, competing with the others well known relationships.

Tables

Table 1. Table 1: Optimization of coefficients using the set A (454 data). New formula.

Group	n	$\sigma$	a	b	c	d	e
e-e	130	0.557	-27.884	-0.952	1.533	-4.101	6.285
e-o	119	0.961	-26.160	-1.140	1.559	17.756	37.055
o-e	109	0.816	-27.800	-0.897	1.535	15.319	30.443
o-o	96	0.915	-24.292	-0.911	1.409	-3.418	7.640

Table 2. Table 2: Optimization of coefficients using the set B.

Group	n	$\sigma$	a	b	c	d	e
e-e	188	0.540	-27.837	-0.94199975	1.5343	-5.7004	8.785
e-o	147	0.678	-28.2245	-0.8629	1.53774	-21.145	53.890
o-e	131	0.522	-26.8005	-1.10783	1.5585	14.8525	-30.523
o-o	114	0.840	-23.6354	-0.891	1.404	-12.4255	36.9005

Table 3. Table 3: Optimization of coefficients using the set C.

Group	n	$\sigma$	a	b	c	d	e
e-e	136	0.298	-26.661	-1.151	1.591	14.680	-50.779
e-o	84	0.914	-32.567	-0.851	1.691	-32.307	111.787
o-e	76	0.859	-30.303	-1.006	1.749	-41.203	124.968
o-o	48	0.810	-25.542	-1.139	1.619	-21.016	109.613

Table 4. Table 4: Parameters of ASAF model; data — the set A.

Group	n	Ci1	Ci2	Ci3	Ci4	Di	y50i
e-e	130	0.985286	0.0179960	0.027056	0.030373	0.001215	-0.018261
e-o	119	1.011020	-0.027134	0.074588	0.051785	0.141828	-0.139752
o-e	109	0.990173	0.063476	0.112658	0.041385	0.087355	-0.168133
o-o	96	0.989577	0.025327	0.226867	0.043133	0.989809	-0.296017

Table 5. Table 5: Parameters of ASAF model for the two other data sets: B, and C.

Group	n	Ci1	Ci2	Ci3	Ci4	Di	y50i
e-e	188 or 136	5.62810	-2.81718	1.53065	-3.97164	5.666667	0.03680
e-o	147 or 84	4.28000	-2.16122	1.55363	-4.07848	3.933333	0.03200
o-e	131 or 76	4.85400	-2.63110	1.56753	-2.92537	4.800000	0.03440
o-o	114 or 48	3.70000	-1.66474	1.46448	-4.60082	2.933333	0.03000

Table 6. Table 6: Standard deviations for ASAF model with the set A, B, and C.

Group	n	${\sigma }_{ASAF}^A$	n	${\sigma }_{ASAF}^B$	n	${\sigma }_{ASAF}^C$
e-e	130	0.731	188	0.415	136	0.438
e-o	119	1.069	147	0.713	84	1.426
o-e	109	1.044	131	0.637	76	1.336
o-o	96	1.041	114	0.876	48	1.069

Table 7. Table 7: Standard deviations of calculated half-lives ( log 10 ⁡ T α ​ ( s ) subscript 10 subscript 𝑇 𝛼 𝑠 \log_{10}T_{\alpha}(s) ) with UNIV after optimization compared to experimental data in four groups of parent nuclei: even-even; even-odd; odd-even, and odd-odd. The set A, B, and C.

Group	n	${\sigma }_{UNIV}^A$	n	${\sigma }_{UNIV}^B$	n	${\sigma }_{UNIV}^C$
e-e	130	0.560	188	0.223	136	0.287
e-o	119	1.050	147	0.533	84	1.384
o-e	109	0.871	131	0.442	76	1.269
o-o	96	0.926	114	0.609	48	1.494

Table 8. Table 8: B k parameters of semFIS formula obtained by fitting the data evaluated by Rytz.

Group	$\sigma$	B1	B2	B3	B4	B5	B6
e-e	0.223	0.993119	-0.004670	0.017010	0.045030	0.018102	-0.025097
o-e	0.533	1.000560	0.010783	0.050671	0.013919	0.043657	-0.079999
e-o	0.442	1.017560	-0.113054	0.019057	0.147320	0.230300	-0.101528
o-o	0.609	1.004470	-0.160560	0.264857	0.212332	0.292664	-0.401158

Table 9. Table 9: B k parameters of semFIS formula obtained by fitting the set C.

Group	$\sigma$	B1	B2	B3	B4	B5	B6
e-e	0.287	0.993119	-0.004670	0.017010	0.045030	0.018102	-0.025097
o-e	1.384	1.017560	-0.113054	0.019057	0.147320	0.230300	-0.101528
e-o	1.269	1.000560	0.010783	0.050671	0.013919	0.043657	-0.079999
o-o	1.494	1.004470	-0.160560	0.264857	0.212332	0.292664	-0.401158

Equations

$${\log }_{10}{T}_{\alpha }(s)=-57.5{\log }_{10}{\mathcal{R}}_{\alpha }(cm)+C$$

$$T_{\alpha }=(100/b_{\alpha })(100/i_p)T_t$$

$$T_{1/2}=a+b{A}^{1/6}\sqrt{Z}+\frac{cZ}{\sqrt{Q_{\alpha }}}$$

$$T_{1/2}=a+b{A}^{1/6}\sqrt{Z}+\frac{cZ}{\sqrt{Q_{\alpha }}}+dI+e{I}^2$$

$$T=[(h\ln 2)/(2E_v)]exp(K_{ov}+K_s)$$

$$K=\frac{2}{\hslash }{\int }_{R_a}^{R_b}\sqrt{2B(R)E(R)}dR$$

$$R_a=R_i+(R_t-R_i)[(E_v+E^{\ast })/E_b^0{]}^{1/2}$$

$$R_b=R_t{E}_c\{1/2+[1/4+(Q+E_v+E^{\ast })E_l/E_c^2{]}^{1/2}\}/(Q+E_v+E^{\ast })$$

$$E_i=E_c+E_l=e^2{Z}_e{Z}_d/R_t+{\hslash }^{2}l(l+1)/(2\mu R_t^2)$$

$$K_{ov}=0.2196(E_b^0{A}_e{A}_d/A{)}^{1/2}(R_t-R_i)\left[\sqrt{1-b^2}-b^2\ln \frac{1+\sqrt{1-b^2}}{b}\right]$$

$$K_s=0.4392[(Q+E_v+E^{\ast })A_e{A}_d/A{]}^{1/2}{R}_b{J}_{rc};b^2=(E_v+E^{\ast })/E_b^0$$

$$J_{rc}$$

$$\lambda =\nu S{P}_s$$

$$\log T=-\log P_s-22.169+0.598(A_e-1)$$

$$-\log P_s=c_{AZ}\left[\arccos \sqrt{r}-\sqrt{r(1-r)}\right]$$

$$\log T=-\log P_s+c_{ee}$$

$$\log T=0.43429K_s\chi -20.446$$

$$K_s$$

$$\chi =B_1+B_{2}y+B_{3}z+B_4{y}^2+B_{5}yz+B_6{z}^2$$

$$y\equiv (N-N_i)/(N_{i+1}-N_i);N_i<N\le N_{i+1}$$

$$z\equiv (Z-Z_i)/(Z_{i+1}-Z_i);Z_i<Z\le Z_{i+1}$$

$$\sigma ={\left\{\sum _{i=1}^n[\log (T_i/T_{exp}){]}^2/(n-1)\right\}}^{1/2}$$

$${\sigma }_{newF}^A=\frac{130{\sigma }_{e-e}+119{\sigma }_{e-o}+109{\sigma }_{o-e}+96{\sigma }_{o-o}}{454}=0.8008$$

$${\sigma }_{newF}^B=\frac{188{\sigma }_{e-e}+147{\sigma }_{e-o}+131{\sigma }_{o-e}+114{\sigma }_{o-o}}{580}=0.6299$$

$${\sigma }_{newF}^C=\frac{136{\sigma }_{e-e}+84{\sigma }_{e-o}+76{\sigma }_{o-e}+48{\sigma }_{o-o}}{344}=0.6438$$

$${\sigma }_{ASAF}^A=\frac{130{\sigma }_{e-e}+119{\sigma }_{e-o}+109{\sigma }_{o-e}+96{\sigma }_{o-o}}{457}=0.9591$$

$${\sigma }_{ASAF}^B=\frac{188{\sigma }_{e-e}+147{\sigma }_{e-o}+109{\sigma }_{o-e}+114{\sigma }_{o-o}}{580}=0.6313$$

$${\sigma }_{ASAF}^C=\frac{136{\sigma }_{e-e}+84{\sigma }_{e-o}+76{\sigma }_{o-e}+48{\sigma }_{o-o}}{344}=0.9657$$

$${\sigma }_{UNIV}^A=\frac{130{\sigma }_{e-e}+119{\sigma }_{e-o}+109{\sigma }_{o-e}+96{\sigma }_{o-o}}{457}=0.8937$$

$${\sigma }_{UNIV}^B=\frac{188{\sigma }_{e-e}+147{\sigma }_{e-o}+131{\sigma }_{o-e}+114{\sigma }_{o-o}}{580}=0.5634$$

Full text

I Introduction

A.H. Becquerel discovered, by chance, $\alpha$, $\beta$, and $\gamma$ radioactivities — the first experimental information coming from an atomic nucleus. E. Rutherford realized that the alpha particle is 4He. H. Geiger and J.M. Nuttal gei11pm gave in 1911 a simple relationship allowing to estimate the half-lives, but only in 1928 the right explanation was given by G. Gamow and simultaneously by R.W. Gurney and E.U. Condon, as a quantum tunnelling through the potential barrier, mostly of electrostatic nature.

As far back as 1911, Geiger and Nuttal have found, for the members of a given natural radioactive family, a simple purely empirical dependence gei11pm of the $\alpha$-decay partial half-life, $T_{\alpha}$, on the mean $\alpha$-particle range, ${\cal{R}}_{\alpha}$, in air (at 15∘ C temperature and one atmosphere pressure), which may be written as:

[TABLE]

where $C$ depends on the series, e.g. $C=41$ for the 238U series. One has approximately ${\cal{R}}_{\alpha}=0.325E_{\alpha}^{3/2}$ in which the kinetic energy of $\alpha$ particles, $E_{\alpha}$, is expressed in MeV and the range in air, ${\cal{R}}$, in cm. This relationship is now of historical interest; the effect of atomic number, $Z$, upon decay rate is obscured. The one-body theory of $\alpha$-decay can explain it and to a good approximation produces a formula with an explicit dependence on the $Z$ number. Nowadays, very often a diagram of $\log T_{\alpha}$ versus $ZQ^{-1/2}$ is called Geiger-Nuttal plot boh75b .

There are many semiempirical relationships (see for example Refs. fro57kd ; wap59b ; taa61b ; vio66jinc ; kel72zp ; hor74np ; hat90pr ; brow92pr ; rur82p ; roy00jpg ; par05app ), allowing to estimate the disintegration period if the kinetic energy of the emitted particle $E_{\alpha}=QA_{d}/A$ is known. $Q$ is the released energy and $A_{d},A$ are the mass numbers of the daughter and parent nuclei. Alpha-decay half-life of an even-even emitter can also be easily calculated by using the universal curves p162jpg91 or the analytical superasymmetric (ASAF) model p195b96b . Some of these formulae were only derived for a limited region of the parent proton and neutron numbers. Their parameters have been determined by fitting a given set of experimental data. Since then, the precision of the measurements was increased and new $\alpha$-emitters have been discovered.

The description of data in the neighborhood of the magic proton and neutron numbers, where the errors of the other relationships are large, was improved by deriving a new formula based on the fission theory of $\alpha$-decay p83jpl80 . A computer program p98cpc82 allows to change automatically the fit parameters, every time a better set of experimental data is available. There are many alpha emitters, particularly in the intermediate mass region, for which both the Q-values and the half-lives are well known wan12cpc ; aud12cpc1 . Initially it was used a set of 376 data (123 even-even (e-e), 111 even-odd (e-o), 83 odd-even (o-e), and 59 odd-odd (o-o)) on the most probable (ground state to ground state or favored transitions) $\alpha$-decays, with a partial decay half-life

[TABLE]

where $b_{\alpha}$ and $i_{p}$, expressed in percent, represent the branching ratio of $\alpha$-decay in competition with all other decay modes, and the intensity of the strongest $\alpha$-transition, respectively.

The formula given by Fröman fro57kd is limited to the region of even-even nuclei with $Z\geq 84$. This formula describes well the experimental data of nuclei with $N\geq 128$ but fails in the region of lighter $\alpha$-emitters, which have not been available at the moment of its derivation. A better overall result gives a simple relationship of Wapstra et al. wap59b also valid for even-even nuclei with $Z\geq 85$. In the new variant derived by A. Brown brow92pr for nuclei with $Z\geq 72$ the agreement with experimental data is not bad for nuclei with $Z\geq 72$, but large errors are obtained for lighter parent nuclei.

The formula presented by Taagepera and Nurmia taa61b remains one of the best. It is exceeded by a variant due to Keller and Münzel kel72zp . Viola and Seaborg vio66jinc introduced a relationship which gives excellent agreement in the region of actinides but it underestimates the lifetimes of lighter nuclei, in contrast with overestimations obtained with the first of the above mentioned formulae.

In the region of superheavy nuclei the majority of researchers prefer to use Viola-Seaborg formula. Very recently for nuclei with $Z=84-110$ and $N=128-160$, for which both $Q_{\alpha}^{exp}$ and $T_{exp}$ experimental values are available, new optimum parameter values par05app have been determined. The average hindrance factors for 45 o-e ($Z=85-107$), 55 e-o ($Z=84-110$), and 40 o-o ($Z=85-111$, $N=129-161$) nuclei were determined to be $C_{V}^{p}=0.437$, $C_{V}^{n}=0.641$, and $C_{V}^{pn}=1.024$. In this way $T_{exp}$ were reproduced by the Viola-Seaborg formula within a factor of 1.4 foe e-e, 2.3 for o-e, 3.7 for e-o and 4.7 for o-o nuclei, respectively. Good results were obtained with a formula due to Royer roy00jpg having 12 parameters $a,b,c$ for e-e, e-o, o-e and o-o nuclei. Shell effects were not taken into account; nuclei with neutron number close to the shell closures $N=152$ and $162$ (namely 3 nuclei with $N=151(Z=96,98,100)$ one with $N=153\ Z=98$, and one with $N=161\ Z=110$) have been omitted in the fitting procedure. Other omission of 3 o-e nuclei with $Z=97,N=146,148$ and $Z=101,N=154$ was motivated by a large deviation from the average behaviour. A simple version of the Viola-Seaborg formula was proposed by Parkhomenko and Sobiczewski par05app .

Since 1979 one of us (DNP) considered $\alpha$ decay a superasymmetric fission process p76jpg79 ; p78jpl79 . Consequently a new semiempirical formula for the alpha decay halflives p83jpl80 was a straightforward finding p83jpl80 ; p101jp83 . Moreover, the analytical and numerical superasymmetric fission (ASAF p103jp84 and NUSAF) models were used together with fragmentation theory developed by the Frankfurt School, and with penetrability calculations like for $\alpha$ decay, to predict cluster (or heavy particle) radioactivity ps84sjpn80 ; enc95 ; p109jpg84 ; p115pr85 ; ps117prl85 ; p309prl11 . The extended calculations p123adnd86 ; p160adnd91 ; p268pr06 have been used to guide the experiments and as a reference for many theoretical developments. A series of books and chapters in books, e.g. p140b89 ; p172b96 ; p195b96 ; p193b97 ; p302bb10 are also available. A computer program p98cpc82 gives us the possibility to improve the parameters of the ASAF model in agreement with a given set of experimental data. The UNIV (universal curve) model was updated in 2011 p308prc11 .

The interest for $\alpha$D is strongly simulated by the search for heavier and heavier superheavies (SHs) — nuclides with $Z>103$, produced by fusion reactions khu14prl ; ham13arnps ; due10prl ; mor07jpsjb who may be identified easily if a chain of $\alpha$D leading to a known nucleus may be measured. Recently it was shown that for superheavy nuclei with atomic numbers $Z>121$ p309prl11 ; p315prc12 $\alpha$D may be stronger than CD or spontaneous fission.

A very interesting result was reported by Y.Z. Wang et al. wan15prc , who compared 18 such formulae in the region of superheavy nuclei. They found: “SemFIS2 formula is the best one to predict the alpha-decay half-lives … In addition, the UNIV2 formula with fewest parameters and the VSS, SP and NRDX formulas with fewer parameters work well in prediction on the SHN alpha-decay half-lives p274el07 ; vio66jinc ; sob07ppnp ; par05app ; ni08prc .”

In this work we intend to study how may be improved a formula developed by G. Royer roy10np by adding few parameters fitted to experimental data. We shall use three data sets, say: A (130 e-e, 119 e-o, 109 o-e, and 96 o-o), set B (188 e-e, 147 e-o, 131 o-e, and 114 o-o), and set C with 136 e-e, 84 e-o, 76 o-e, and 48 o-o alpha emitters.set C with 136 e-e, 84 e-o, 76 o-e, and 48 o-o alpha emitters. The set A was developed by one of us (DA), the set B belongs to DNP’s group, and the set C was taken from G. Royer roy10np ; few Q-values have been updated using the AME12 evaluation of experimental atomic masses wan12cpc . Comparison with ASAF, UNIV, and semFIS will be made using both A, B and C data sets.

II New formula

The Royer formula roy10np is defined as

[TABLE]

with initial parameters $a=-27.657;-28.408;-27.408,and-24.763$, $b=-0.966;-0.920;$ $-1.038,and-0.907$, and $c=1.522;1.519;1.581,and1.410$ for e-e, e-o, o-e, and o-o, respectively. The rms standard deviation for 130 e-e, 119 e-o, 109 o-e, and 96 o-o was $\sigma=0.560,1.050,0.871,$ and $0.926$, respectively.

The new relationship is obtained by introducing $I=(N-Z)/A$ and the new parameters d and e:

[TABLE]

where initially for the set A the parameters a, b, c, d, e are given in table 1.

Before optimization, with our set of 580 $\alpha$ emitters, and the initial values of the parameters $a=-27.989,b=-0.940,c=1.532,d=-5.747,e=11.336$ for even-even nuclei we got the following values of rms standard deviations, $\sigma=0.5547$. After optimization for e-e emitters, with $a=-27.837,b=-0.94199975,c=1.5343,d=-5.7004,e=8.785$ the agreement was improved: $\sigma=0.540$. The order of optimization of the 5 parameters was: a; e; d; c, and b.

We may compare the results obtained by using the set A, B and C in tables 1 and 2. We can see a slight improvement by using the set B.

III ASAF

The half-life of a parent nucleus $AZ$ against the split into a cluster $A_{e}Z_{e}$ and a daughter $A_{d}Z_{d}$

[TABLE]

is calculated by using the WKB quasiclassical approximation, according to which the action integral is expressed as

[TABLE]

with $B=\mu$ — the reduced mass, $K=K_{ov}+K_{s}$, and $E(R)$ replaced by $[E(R)-E_{corr}]-Q$. $E_{corr}$ is a correction energy similar to the Strutinsky str67np shell correction, also taking into account the fact that Myers-Swiatecki’s liquid drop model (LDM) mye66np overestimates fission barrier heights, and the effective inertia in the overlapping region is different from the reduced mass. The turning points of the WKB integral are:

[TABLE]

[TABLE]

where $E^{*}$ is the excitation energy concentrated in the separation degree of freedom, $R_{i}=R_{0}-R_{e}$ is the initial separation distance, $R_{t}=R_{e}+R_{d}$ is the touching point separation distance, $R_{j}=r_{0}A_{j}^{1/3}$ $(j=0,e,d;\;r_{0}=1.2249$ fm) are the radii of parent, emitted and daughter nuclei, and $E_{b}^{0}=E_{i}-Q$ is the barrier height before correction. The interaction energy at the top of the barrier, in the presence of a non negligible angular momentum, $l\hbar$, is given by:

[TABLE]

The two terms of the action integral $K$, corresponding to the overlapping ($K_{ov}$) and separated ($K_{s}$) fragments, are calculated by analytical formulas (approximated for $K_{ov}$ and exact for $K_{s}$ in case of separated spherical shapes within the LDM):

[TABLE]

[TABLE]

[TABLE]

where $r=R_{t}/R_{b}$ and $c=rE_{c}/(Q+E_{v}+E^{*})$. In the absence of the centrifugal contribution ($l=0$), one has $c=1$.

The choice $E_{v}=E_{corr}$ allows to get a smaller number of parameters. It is evident that, owing to the exponential dependence, any small variation of $E_{corr}$ induces a large change of T, and thus plays a more important role compared to the preexponential factor variation due to $E_{v}$. Shell and pairing effects are included in $E_{corr}=a_{i}(A_{e})Q$ ($i=1,2,3,4$ for even-even, odd-even, even-odd, and odd-odd parent nuclei). For a given cluster radioactivity there are four values of the coefficients $a_{i}$, the largest for even-even parent and the smallest for the odd-odd one (see figure 1 of p160adnd91 ). The shell effects for every cluster radioactivity is implicitly contained in the correction energy due to its proportionality with the Q value, which is maximum when the daughter nucleus has a magic number of neutrons and protons.

With only few exceptions, in the region of nuclei far from stability, measured $\alpha$-decay partial half-lives are not available. In principle we can use the ASAF model to estimate these quantities. Nevertheless, slightly better results can be obtained by using semFIS p195b96a . The potential barrier shape similar to that we considered within the ASAF model was calculated by using the macroscopic-microscopic method p266pr06 , as a cut through the PES at a given mass asymmetry, usually the 208Pb valley or not far from it.

Before any other model was published, there were estimations of the half-lives for more than 150 decay modes, including all cases experimentally confirmed until now. A comprehensive table was produced by performing calculations within that model. Subsequently, the numerical predictions of the ASAF model have been improved by taking better account of the pairing effect in the correction energy, deduced from systematics in four groups of parent nuclei (even - even, odd - even, even - odd and odd - odd). In a new table, published in 1986, cold fission as cluster emission has been included. The systematics was extended in the region of heavier emitted clusters (mass numbers A${}_{e}>$ 24), and of parent nuclei far from stability and superheavies. Since 1984, the ASAF model results have been used to guide the experiments and to stimulate other theoretical works.

IV UNIV (Universal Formula)

In cluster radioactivity and $\alpha$-decay the (measurable) decay constant $\lambda=\ln 2/T$, can be expressed as a product of three (model dependent) quantities

[TABLE]

where $\nu$ is the frequency of assaults on the barrier per second, $S$ is the preformation probability of the cluster at the nuclear surface, and $P_{s}$ is the quantum penetrability of the external potential barrier. The frequency $\nu$ remains practically constant, the preformation differs from one decay mode to another but it is not changed very much for a given radioactivity, while the general trend of penetrability follows closely that of the half-life. The external part of the barrier (for separated fragments), essentially of Coulomb nature, is much wider than the internal one (still overlapping fragments).

According to Ref. p167ps91 the preformation probability can be calculated within a fission model as a penetrability of the internal part of the barrier, which corresponds to still overlapping fragments. One may assume as a first approximation, that preformation probability only depends on the mass number of the emitted cluster, $S=S(A_{e})$. The next assumption is that $\nu(A_{e},Z_{e},A_{d},Z_{d})=$ constant. In this way one arrives at a single straight line universal curve on a double logarithmic scale

[TABLE]

where

[TABLE]

with $c_{AZ}=0.22873({\mu}_{A}Z_{d}Z_{e}R_{b})^{1/2}$, $r=R_{t}/R_{b}$, $R_{t}=1.2249(A_{d}^{1/3}+A_{e}^{1/3})$, $R_{b}=1.43998Z_{d}Z_{e}/Q$, and $\mu_{A}=A_{d}A_{e}/A$.

Sometimes this universal curve is misinterpreted as being a Geiger-Nuttal plot. Nowadays by Geiger-Nuttal diagram one understands a plot of $\log T$ versus $ZQ^{-1/2}$, or versus $Q^{-1/2}$. In this kind of systematics the experimental points are scattered. Nevertheless, for a given atomic number, $Z$, or for the members of a natural radioactive series, it is still possible to get a single straight line.

The strong shell effect at the magic neutron number $N=126$, which was ignored when the approximation $S=S(A_{e})$ was made to give a pronounced underestimation of the half-lives in the neighborhood of $N=126$.

For $\alpha$-decay of even-even nuclei, $A_{e}=4$, one has

[TABLE]

where $c_{ee}=\log S_{\alpha}-\log\nu+\log(\ln 2)=-20.375$. We can find new values for $c_{ee}$ and we also can extend the relationship to even-odd, odd-even, and odd-odd nuclei, by fitting a given set of experimentally determined alpha decay data.

V semFIS (Semiempirical relationship based on fission theory of

$\alpha$-decay)

Mainly the $Z$ dependence was stressed by all formulae, in spite of strong influence of the neutron shell effects. The neighborhood of the magic numbers of nucleons is badly described by all these relationships.

The SemFIS formula based on the fission theory of $\alpha$-decay gives

[TABLE]

where

[TABLE]

and the numerical coefficient $\chi$, close to unity, is a second-order polynomial

[TABLE]

in the reduced variables $y$ and $z$, expressing the distance from the closest magic-plus-one neutron and proton numbers $N_{i}$ and $Z_{i}$:

[TABLE]

[TABLE]

with $N_{i}=....,51,83,127,185,229,.....$ , $Z_{i}=....,29,51,83,115,.....$ , and $Z_{da}=Z-2$ , $A_{da}=A-4$ . The coefficients $B_{i}$ obtained by using a high-quality selected set of alpha-decay data ryt91adnd are given in the Table 8. Better agreement with experimental results are obtained in the region of superheavy nuclei by introducing other values of the magic numbers plus one unit for protons (suggesting that the next magic number of protons could be 126 instead of 114): $Z_{i}=....,83,127,165,.....$

With the SemFIS formula, Rurarz rur82p have made predictions for nuclei far from stability with $62<Z<76$. In the variant of Ref. hat90pr the shell effect on the formation factor was approximated by an empirical relationship.

Practically for even-even nuclei, the increased errors in the neighborhood of $N=126$, present in all other cases, are smoothed out by SemFIS formula using the second order polynomial approximation for $\chi$. They are still present for the strongest $\alpha$-decays of some even-odd and odd-odd parent nuclides. In fact for non-even number of nucleons the structure effects became very important, and they should be carefully taken into account for every nucleus, not only globally. An overall estimation of the accuracy, gives the standard rms deviation of $\log T$ values:

[TABLE]

The parameters $\{B_{k}\}$ of the SemFIS formula could be automatically improved, for a given set of experimental data, by using the computer program described in the Ref. p98cpc82 . The partial $\alpha$-decay half-lives plotted in this figure are lying in the range of $10^{-7}$ to $10^{25}$ seconds. One can see the effect of the spherical and deformed neutron magic numbers of the daughter nuclei $N_{d}=126,152,162$ particularly clear for even-even and even-odd nuclides. For the large set of alpha emitters the following values of the rms errors have been obtained: $\log T$: 0.19 for SemFIS formula; 0.33 for the universal curve; 0.39 for ASAF model, and 0.43 for numerical superasymmetric (NuSAF) model p195b96b .

There are many parameters of the SemFIS formula introduced in order to reproduce the experimental behaviour around the magic numbers of protons and neutrons, which could be a drawback in the region of light and intermediate alpha emitters. In the region of superheavies these characteristics may be conveniently used to get informations concerning the next magic numbers of protons and neutrons which are not well known until now. When accurate experimental values of $Q$ and $T$ are available in the region centered on $Z=114-126$, $N=172-184$, the SemFIS formula may be used to estimate whether the right value of the spherical magic number is $Z=114$, $Z=120$, $Z=126$, and $N=172$, or $N=184$, due to the high sensitivity of $\chi$ to the values of $Z_{i}$ and $N_{i}$ (see eqs. 17–21).

In figure 2 we plotted the individual errors: differences of $\log_{10}T_{theor}-\log_{10}T_{exp}$ in four groups of alpha emitters. Vertical dashed lines are marking magic numbers of neutrons, either spherical (50, 82, 126) or deformed (152, 162, 172). One can see that the best result is always obtained for even-even alpha emitters.

Superheavy (SH) nuclei, with atomic numbers $Z=104-118$, are decaying mainly by $\alpha$ decay and spontaneous fission. They have been produced in cold fusion or hot fusion (48Ca projectile) reactions khu14prl ; ham13arnps ; due10prl ; mor07jpsjb . In a systematic study of $\alpha$-decay energies and half-lives of superheavy nuclei it was shown wan15prc that our semFIS (semiempirical formula based on fission theory) and UNIV (universal curve) are the best among 18 calculations methods of $\alpha$ decay half-lives. For some isotopes of even heavier SHs, with $Z>121$, there is a good chance for cluster decay modes to compete p315prc12 ; p309prl11 .

VI Comparison of results obtained with the new formula, semFIS, UNIV,

and ASAF.

We present in figure 3 the results obtained using the sets A, B, and C, respectively. A global indicator for a given model could be the weighted mean value

[TABLE]

[TABLE]

[TABLE]

Similarly for the other models

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

From these results we may say that globally semFIS, with $\sigma_{semFIS}^{B}=0.4269$, is the best one, followed in order by UNIV, $\sigma_{UNIV}^{B}=0.5634$, ASAF, $\sigma_{newF}^{B}=0.6299$, and $\sigma_{ASAF}^{B}=0.6313$. Nevertheless, it is interesting to observe that for even-even alpha emitters newF gives the 3rd best result after semFIS ($\sigma_{newFe-e}=0.298$ compared to $\sigma_{semFISe-e}^{B}=0.223$ and $\sigma_{semFISe-e}^{C}=0.287$), and for odd-even parents the 2nd is newF with $\sigma_{newFo-e}^{B}=0.522$ compared to $\sigma_{semFISo-e}=0.442$. newF is also better than ASAF for o-o nuclides, when $\sigma_{newFo-o}^{B}=0.840$ and $\sigma_{ASAFo-o}^{B}=0.876$.

VII Conclusions

The accuracy of the new formula was increased after optimization of the five parameters in the order: a; e; d; c, and b. The SemFIS formula taking into account the magic numbers of nucleons, the analytical super-asymmetric fission model and the universal curves may be used to estimate the alpha emitter half-lives in the region of superheavy nuclei. The dependence on the proton and neutron magic numbers of the semiempirical formula may be exploited to obtain informations about the values of the magic numbers which are not well known until now.

We introduced a weighted mean value of the rms standard deviation, allowing to compare the global properties of a given model. In this respect for the set B the order of the four models is the following: semFIS; UNIV; newF, and ASAF. Nevertheless for even-even alpha emitters UNIV gives the 2nd best result after semFIS, and for odd-even parents the 2nd is newF.

The quality of experimental data was also tested, as one can see by comparing the three sets (A, B, C). The set B with large number of emitters (580) gives the best global result. It is followed by the set A (454) three times and the set C (344).

Despite its simplicity in comparison with semFIS the new formula, presented in this article, behaves quite well, competing with the others well known relationships discussed in the Ref.  wan15prc .

Acknowledgements.

This work was supported within the IDEI Programme under Contracts No. 43/05.10.2011 and 42/05.10.2011 with UEFISCDI, and NUCLEU Programme PN16420101/2016 Bucharest.