Diffuseness parameter as a bottleneck for accurate half-life calculations
DDiffuseness parameter as a bottleneck for accurate half-life calculations
Aladdin Abdul-latif
1, 2 and Omar Nagib ∗ Department of Physics, German University in Cairo, Cairo 11835, Egypt Department of Physics, Cairo University, Giza 12613, Egypt (Dated: January 19, 2021)An investigation of the calculated α decay half-lives of super heavy nuclei (SHN) reveals that the diffusenessparameter is a great bottleneck for achieving accurate results and predictions. In particular, when universal prox-imity function is adopted for nuclear potential, half-life is found to vary significantly and nonlinearly as a func-tion of diffuseness parameter. To overcome this limiting hurdle, a new semiempirical formula for diffusenessthat is dependent on charge and neutron numbers is proposed in this work. With the model at hand, half-lives of218 SHN are computed, for 68 of which there exists available experimental data and 150 of which are predicted.The calculations of half-lives for 68 SHN are compared against experimental data and the calculated data ob-tained by using deformed Woods-Saxon, deformed Coulomb potentials model, and six semiempirical formulas.The predictions of 150 SHN are compared against the predictions of seven of the current best semiempirical for-mulas. Calculations of the present study are in good agreement with the experimental half-lives outperformingall but ImSahu semiempirical formula. Moreover, the predictions of our model are consistent with predictionsof the semiempirical formulas. We strongly conclude that more attention should be directed toward obtainingaccurate diffuseness parameter values for using it in nuclear calculations. I. INTRODUCTION
Currently, within the theoretical framework of nuclearphysics, especially in α decay of super heavy nuclei (SHN),there are points of intersection and divergence between dif-ferent models used in calculations. Particularly, points of dis-agreement abound, including the particular functional formof nuclear interaction between α /cluster and the daughter nu-clei, be it given by Woods-Saxon (WS) potential [1], doublefolding model (DFM) [2, 3], liquid drop model (LDM) [4],universal proximity potential [5, 6] and et cetera. Anotherpoint of disagreement stems from two competing proposalsfor α decay. The first–often dubbed cluster-like theory– as-signs a preformation factor P o for every parent nuclei, whilethe second fission-like model asserts that P o = 1 for all nuclei[7]. On the other hand, the diffuseness parameter a is one ofthe parameters with universal presence in all models regard-less of the particular nuclear potential under study. Althoughubiquitous, little attention is paid to the parameter which isoften set constant for all nuclei [1, 8–11]. More recently, De-hghani et al. investigated the role diffuseness parameter playswhen deformed Woods-Saxon potential is adopted for nuclearinteraction between α and daughter nuclei. Some interestingconclusions were reached of which we list the chief ones [12]:1. The logarithm of half-life log T / decreases linearlywith increasing diffuseness parameter. For instance, avariation of 0.4 fm of a can induce a change in the log-arithm of half-life by 2 (i.e., half-life changes by twoorders of magnitude).2. For a given nucleus, a systematic search was carried outin search for the best a value that matches the exper-imental half-life for that particular nucleus. This pro-cess was applied on 68 SHN and diffuseness parameter ∗ [email protected] for 68 SHN was extracted out of experimental data ofhalf-lives.3. Adopting a constant a = 0 . fm for all nuclei wasfound to be optimal in the sense of minimizing the root-mean-square (rms) error of the logarithm of half-lives.The rms error came out to be 0.787, implying that onaverage, adopting a = 0 . fm will induce an error of0.787 for the logarithm of half-life.Our work extends and generalizes these results by investigat-ing the effect of diffuseness parameter when universal prox-imity potential is adopted for nuclear interaction between α and daughter nuclei. The outline of this paper is as follows: inSec. II, we outline the theoretical framework that we will beworking with regarding calculations of half-lives. In Sec. III,the relation between diffuseness and half-life is probed whenproximity potential is adopted and results are compared withthe case of deformed WS, deformed Coulomb potentials. InSec. IV, we investigate the relation between diffuseness pa-rameter on one hand and charge, neutron and mass numberson the other, and we propose a new semiempirical formula fordiffuseness as a function of charge and neutron numbers. InSec. V, calculations and predictions of 218 SHN are made;calculations for 68 SHN will be carried out using proxim-ity potential with variable effective diffuseness and comparedwith calculations of deformed WS, deformed Coulomb po-tentials in which diffuseness parameter is set to a = 0 . fm.Moreover, our calculations will be compared with six popularsemiempirical formulas for half-lives. The viability of the newformula for diffuseness is tested by predicting the half-lives of150 SHN and comparing the results with seven of the currentbest semiempirical formulas. We conclude the last section bydiscussing the most important highlights, results, and poten-tial research directions in the future. a r X i v : . [ nu c l - t h ] J a n II. THEORETICAL FRAMEWORK
Generally, the effective potential for α decay can be bro-ken into three contributions, namely nuclear, Coulomb, andangular contributions V eff ( r ) = V N ( r ) + V C ( r ) + V l ( r ) (1)where r is the separation distance between the center of thedaughter and α nuclei. For our particular potential, we assumespherical symmetry and ignore deformation effects, i.e., weconsider the potential to assume radial dependence only, withthe quadrupole and hexadecapole deformation parameters β and β set to zero. Moreover, we shall consider cases in whichthere’s no total angular momentum carried by the α -daughtersystem and hence V l ( r ) is subsequently ignored. Regardingthe nuclear term V N ( r ) , we adopt the universal proximity po-tential previously proposed by Zhang et al. [5]: V N ( s ) = 4 πb eff Rγφ ( s ) (2) φ ( s ) = p (cid:16) s + p p (cid:17) (3)In two parameter Fermi distribution (2pF) of nuclear matter, b eff and the effective diffuseness a eff of the proximity potentialare related as b eff = π √ a eff (4)The effective diffuseness is a function of both the nuclearsurface diffuseness of the daughter and α nuclei. Followingprevious works [5, 13], we posit the ansatz that a eff is the av-erage of the two, i.e., a eff = ( a α + a d ) / with a α = 0 . fmand a d refers to the nuclear surface diffuseness of the daugh-ter. We shall explain and show hereafter that adopting a con-stant a eff for all nuclei leads to unacceptable errors, especiallyfor the proximity potential and for systems in which we ex-pect that a eff ≥ . fm (or equivalently b eff ≥ . fm). γ isgiven by γ = 0 . (cid:104) − . (cid:16) N − ZA (cid:17) (cid:105) MeV fm − (5) R = R α R d / ( R α + R d ) is the reduced radius of the α -daughter system where the radius of each nucleus in terms ofits mass number is given by R = 1 . A / + 0 . A − / − . (6) φ ( s ) is the universal function that quantifies the nuclearinteraction between α and the daughter in terms of the reducedseparation distance s s = r − R α − R d b eff (7)In particular, this equation holds in the regime s > − .The constants p , p and p appearing in Eq. (3) are given by-7.65, 1.02 and 0.89 respectively. Moving on, the Coulombterm V C ( r ) is given by V C ( r ) = Z α Z d e r r ≥ R C R C (cid:104) − (cid:16) rR C (cid:17) (cid:105) r < R C (8)where Z α and Z d are the α and daughter charge numbersand R C = R α + R d . In the context of WKB approximation,the half-life is given by T / = π (cid:126) ln 2 P o E ν (cid:16) K ) (cid:17) (9)where P o , E ν and K are the preformation factor, zero-pointvibration energy and action integral respectively. The actionintegral is given by K = 2 (cid:126) (cid:90) r r (cid:112) µ ( V eff ( r ) − Q α ) dr (10)where Q α , µ, r , and r are the decay energy, reducedmass, and first and second turning points, respectively. Weshall describe E ν classically, since it was previously found[7] that classical and quantum mechanical (e.g., modified har-monic oscillator) approaches yield not too different results,hence E ν = (cid:126) ω (cid:126) π R p (cid:114) E α m α (11)where R p , E α and m α are the parent radius, kinetic energyof α nuclei and its mass respectively. The decay energy ofthe system and the kinetic energy of α nuclei are related bythe recoil of the daughter and electron shielding, or in otherwords [14], E α = A d A p Q α − (cid:16) . Z / d − Z / d (cid:17) · − MeV (12)where A d and A p are the mass numbers of the daughter andthe parent nuclei respectively. Finally, for the preformationfactor, we adopt [15] log P o = a + b ( Z − Z )( Z − Z )+ c ( N − N )( N − N )+ dA + e ( Z − Z )( N − N ) (13)where a, b, c, d and e are given by 34.90593, 0.003011,0.003717, -0.151216 and 0.006681 respectively. For ≤ Z ≤ and ≤ N ≤ , which is the scope ofthe present paper, the magic numbers Z , Z , N and N aregiven by 82, 126, 152 and 184 respectively. III. DIFFUSENESS AND HALF-LIFE FOR PROXIMITYPOTENTIAL
A priori , one should not expect the effective diffuseness pa-rameter a eff for the proximity potential and the diffuseness pa-rameter for WS potential a WS to be equal or even play the samelogical roles; this is owing to the different functional forms ofthe two potentials. To be more precise, consider the unde-formed WS potential [1] V WS ( r ) = − V (cid:16) r − R a WS (cid:17) (14)where V is the depth of the potential well and R is thehalf radius at which V ( R ) = − V / . One can easily see thatletting a WS → ∞ leads to V WS ( r ) → − V / for all r , while a WS → leads to V WS ( r ) → − V for r ≤ R . In other words,smaller a WS leads to a deeper potential well (and vice versa)with the depth of the potential varying from − V to − V / .Regarding the proximity potential, however, from Eqs. (2),(3) and (7), we see that a eff → ∞ leads to V N ( r ) → −∞ for all r , while a eff → leads to V N ( r ) → for all r ; thatis, greater a eff leads to a deeper potential well (unlike the WScase) and vice versa, with the depth of the potential varyingfrom −∞ to 0.FIG. 1. log T / vs. a eff (fm) for 256Rf.This is the motivation behind the current investigation. Inthis section, we shall study how half-lives of nuclei vary with a eff . Simulation results displayed in Fig. 1 and 2 show vari-ation of the logarithm of half-life with a eff as it’s varied from0.22 to 0.61 fm. Simulation results are interesting insofar thatthey intersect and diverge from the previous work which in-vestigated the role of diffuseness [12]. Both results confirmthe physical intuition that half-life is a monotonically decreas-ing function of diffuseness parameter; this is to be expectedsince increasing a eff increases the range of nuclear interac-tion R which dominates the Coulomb force, leading to a de-creased potential height and hence increased penetrability ofthe barrier and a decrease in half-life. More interesting per-haps is the different behavior exhibited by half-life when prox-imity potential is adopted instead of deformed WS, deformedCoulomb potentials as in the previous study [12]. Unlike the FIG. 2. log T / vs. a eff (fm) for three different SHN.WS case–in which half-life varies linearly with diffusenessparameter– the half-life here is a non-linear convex functionof the effective diffuseness a eff . In Fig. 1, we can see that vari-ation of diffuseness from 0.22 to 0.61 fm leads to change in thelogarithm of half-life from 2.1 to -0.1– two orders of magni-tude change in the half-life (equivalently, b eff was varied from0.4 to 1.1 fm). In addition to the significant change in half-life with small variations in diffuseness, the non-linear convexbehavior implies that for systems with a eff > . fm, approx-imating diffuseness as 0.54 fm is problematic since half-lifevaries even stronger with diffuseness compared to the region a eff < . fm; to illustrate with an example, consider the caseof 256Rf shown in Fig. 1, varying a eff by 0.1 fm from 0.22 to0.32 fm induces a change of 0.457 in the logarithm of half-life,while varying a eff from 0.5 to 0.6 fm leads to a change in thehalf-life by 0.69. These two aforementioned facts– significantchange with slight diffuseness variation and convexity/non-linearity– deem common approximations [5, 16–20] such as b eff = 0 . − fm unacceptable since they’re bound to pro-duce large errors. Fig. 2 helps to illustrate the universality ofthis behavior irrespective of the SHN under study. IV. NEW SEMIEMPIRICAL FORMULA FOR a eff In their work, Dehghani et al. considered deformed WS,deformed Coulomb potentials and investigated 68 SHN in theregion ≤ Z ≤ [12]. For every nucleus, a system-atic search was carried out looking for the optimal a WS valuethat matches the experimental half-life for that particular nu-cleus. The result of their work is plotted in Fig. 3 in whichdiffuseness a WS is plotted against Z, N , and A . We see that a WS can be as large as 0.86 fm for the particular nucleus understudy. Moreover, the general trend is an increase of a WS when Z and N (or equivalently A ) is increased although this is nota strict rule. To map a WS into a eff , we posit the ansatz that a eff = ( a α + a WS ) / ; the reasonableness and viability of thisassumption will be addressed by the results in the followingsection. By mapping a WS into a eff , we get the plot in Fig. 4 foreffective diffuseness vs. Z, N , and A . Next, we tried fittingboth a WS and a eff vs. Z and N . Various fitting schemes includ-ing linear and non-linear ones were considered and tried. Welist the two main results of these attempts:FIG. 3. a WS (fm) vs Z, N , and A .FIG. 4. a eff (fm) vs Z, N , and A .1. For a eff , a linear fit best represents the data with an rmserror of 0.065 fm from true a eff values. The obtainedsemiempirical formula is given by a eff = − . . Z + 0 . N (15)2. No fitting model was found for a WS that approximatetrue a WS with a reasonable rms error comparable to theone for a eff . In particular, linear and non-linear mod-els produced rms error of around 0.13-0.16 fm whereasadopting constant a WS = 0 . fm produced rms error ofaround 0.17 fm.From comparing the figures, one can understand why it’seasier to find a fit for a eff in which points are condensed moretightly compared to the more scattered and disperse data for a WS . Moreover, there’s a stronger Z dependence in both casescompared to N dependence, which is reflected in the fact thatthe coefficient of Z is one order of magnitude larger than thatof N . V. CALCULATIONS AND PREDICTIONS OF α HALF-LIVES OF 218 SHN
In this section, we calculate and predict α half-lives of 218SHN based on the theoretical framework outlined earlier andthe semiempirical formula proposed. In particular, we calcu-late half-lives for 68 SHN for which there exists available ex-perimental data using our proximity potential model; for thesenuclei we do not use the semiempirical formula but rathercompute a eff directly from a WS using the mapping outlined inthe previous section; moreover, the results obtained are com-pared against previous work that assumed deformed WS, de-formed Coulomb potentials and constant a WS = 0 . fm forall nuclei (WS model) [12]. Moreover, we compare our out-put with the results of six of the most powerful semiempiricalformulas. Regarding the 150 SHN whose half-lives to be pre-dicted, there’s no available a WS for them and hence we makeuse of our proposed semiempirical formula to compute a eff values. Experimental or theoretical Q α values are taken fromRefs. [12, 14, 21].The results of the calculations for the 68 SHN are tabu-lated in Table I, where Z varies from 104 to 118 and N variesfrom 152 to 176. The table also contains the logarithm oferrors for our model ∆ calc = log T exp − log T calc and ∆ deformed WS = log T exp − log T deformed WS for the deformed WS,deformed Coulomb model with constant diffuseness. More-over, a plot of the errors of the two models vs. Z is shown inFig. 5. We note that one calculation is not included in the ta-ble, graphs or earlier fitting procedure, namely, that of 256Dbwith error ∆ > for both models. We worked with arbitraryprecision in our calculations but rounded up numbers in the ta-bles to the nearest 2 or 3 decimal places. Fig. 5 is instructiveto draw conclusions from; in particular, it shows our model–dubbed Prox– performs favorably in several aspects. First, isthat most of the points are clustered around the 0.5 to -0.5band, unlike the WS model which has more disperse pointsand are equally likely to be found inside and outside the 0.5band. Second, | ∆ calc | < for all points in our model (except256Db which was mentioned earlier); this is to be contrastedwith the results of WS model in which there are more than21 points exceeding the 1 to -1 band (i.e., | ∆ deformed WS | > )with 3 points even exceeding 1.5 or -1.5. Statistical compari-son between the performance of our model (Prox), WS model,and six semiempirical formulas (ImSahu, Sahu, Royer10, VS,SemFIS and UNIV) [21] is shown in Table II in which rms,mean, mean deviation and difference between the maximumand minimum errors are shown respectively. The rms of errorsis given by (cid:112) δ , (cid:112) δ = (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) M M (cid:88) i ∆ i (16)where M is the number of SHN under study. The mean oferrors is simply given byTABLE I. Experimental data log T exp vs. our model Prox log T calc with variable effective diffuseness a eff vs. log T deformed WS deformed Woods-Saxon, deformed Coulomb potentials with constant diffuseness a WS = 0 . fm. Values for Q α and log T deformed WS are from Ref. [12]. Z N A Q α a eff b eff log T exp log T calc log T deformed WS ∆ calc ∆ deformed WS
104 152 256 8.926 0.42 0.77 0.319 1.093 0.265 -0.770 0.054104 154 258 9.190 0.49 0.89 -1.035 -0.398 -0.512 -0.645 -0.523104 159 263 8.250 0.35 0.64 3.301 2.997 2.592 0.311 0.709105 152 257 9.206 0.37 0.67 0.389 1.039 -0.169 -0.649 0.558105 153 258 9.500 0.22 0.40 0.776 0.777 -1.036 -0.001 1.812105 154 259 9.620 0.31 0.57 -0.292 -0.101 -1.341 -0.192 1.049105 158 263 8.830 0.35 0.64 1.798 1.619 1.056 0.152 0.715106 153 259 9.804 0.32 0.58 -0.492 -0.043 -1.482 -0.449 0.990106 154 260 9.901 0.42 0.78 -1.686 -1.001 -1.759 -0.685 0.073106 155 261 9.714 0.36 0.66 -0.638 -0.268 -1.222 -0.37 0.584106 156 262 9.600 0.49 0.89 -1.504 -0.855 -0.921 -0.648 0.583106 163 269 8.700 0.41 0.75 2.079 1.785 1.913 0.294 0.166106 165 271 8.670 0.41 0.75 2.219 1.775 1.996 0.444 0.223107 153 260 10.40 0.29 0.53 -1.459 -0.969 -2.698 -0.490 1.239107 154 261 10.50 0.31 0.57 -1.899 -1.459 -2.916 -0.440 1.017107 157 264 9.960 0.30 0.55 -0.357 -0.390 -1.479 0.033 1.122107 159 266 9.430 0.41 0.75 0.23 0.290 0.005 -0.060 0.225107 160 267 9.230 0.37 0.67 1.23 1.024 0.643 0.206 -0.413107 163 270 9.060 0.36 0.66 1.785 1.364 1.140 0.421 0.645107 165 272 9.310 0.36 0.66 1.000 0.474 0.356 0.526 0.644107 167 274 8.930 0.42 0.77 1.732 1.218 1.598 0.515 0.134108 156 264 10.591 0.44 0.80 -2.796 -2.219 -2.750 -0.577 -0.046108 157 265 10.47 0.46 0.84 -2.699 -2.167 -2.464 -0.532 -0.235108 158 266 10.346 0.48 0.87 -2.638 -2.097 -2.141 -0.54 -0.497108 159 267 10.037 0.41 0.75 -1.187 -0.956 -1.344 -0.23 0.157108 162 270 9.050 0.54 0.98 0.556 0.810 1.55 -0.254 -0.994108 165 273 9.730 0.39 0.71 -0.119 -0.490 -0.506 0.37 0.387109 159 268 10.67 0.33 0.60 -1.678 -1.684 -2.613 0.006 0.935109 165 274 10.20 0.32 0.58 -0.357 -0.973 -1.394 0.616 1.037109 166 275 10.48 0.38 0.69 -1.699 -2.086 -2.127 0.386 0.428109 167 276 10.03 0.36 0.66 -0.347 -0.848 -0.966 0.501 0.619109 169 278 9.580 0.40 0.73 0.653 0.106 0.358 0.547 0.295110 157 267 11.78 0.51 0.93 -5.553 -4.625 -4.811 -0.926 -0.742110 159 269 11.509 0.38 0.69 -3.747 -3.472 -4.242 -0.275 0.495110 160 270 11.12 0.50 0.91 -4.00 -3.455 -3.327 -0.545 -0.673110 161 271 10.899 0.41 0.75 -2.639 -2.465 -2.808 -0.174 0.169110 163 273 11.38 0.41 0.75 -3.77 -3.894 -3.927 0.124 0.157110 167 277 10.72 0.41 0.75 -2.222 -2.518 -2.413 0.295 0.191110 171 281 9.32 0.37 0.67 2.125 1.403 1.596 0.723 0.529111 161 272 11.197 0.34 0.62 -2.420 -2.338 -3.193 -0.080 0.773111 163 274 11.48 0.24 0.44 -2.194 -2.7027 -3.864 0.508 1.67111 167 278 10.85 0.43 0.78 -2.377 -2.557 -2.415 0.180 0.038111 168 279 10.53 0.37 0.67 -1.046 -1.465 -1.578 0.419 0.532111 169 280 9.91 0.37 0.67 0.663 0.151 0.111 0.511 0.552112 169 281 10.46 0.43 0.78 -1.000 -1.288 -1.040 0.287 0.040112 173 285 9.32 0.53 0.96 1.447 1.175 2.424 0.2715 -0.977113 165 278 11.85 0.38 0.69 -3.62 -3.589 -4.053 -0.032 0.433113 169 282 10.78 0.42 0.73 -1.155 -1.525 -1.437 0.370 0.282113 170 283 10.48 0.47 0.86 -1.00 -0.812 -0.640 -0.189 -0.360113 171 284 10.12 0.48 0.87 -0.041 -0.382 0.442 0.340 0.483113 172 285 10.01 0.44 0.80 0.623 0.137 0.738 0.486 -0.115113 173 286 9.79 0.48 0.87 0.978 0.482 1.431 0.490 -0.453114 172 286 10.35 0.53 0.96 -0.699 -0.994 0.273 0.294 -0.972114 173 287 10.17 0.54 0.98 -0.319 -0.610 0.755 0.290 -1.074114 174 288 10.072 0.55 1.00 -0.180 -0.440 1.018 0.259 -1.198
TABLE I. (
Continued ). Z N A Q α a eff b eff log T exp log T calc log T deformed WS ∆ calc ∆ deformed WS
114 175 289 9.98 0.53 0.97 0.279 -0.066 1.264 0.345 -0.985115 172 287 10.76 0.53 0.97 -1.432 -1.682 -0.451 0.249 -0.981115 173 288 10.63 0.53 0.97 -1.060 -1.393 -0.121 0.333 -0.939115 174 289 10.52 0.51 0.93 -0.658 -1.010 0.156 0.342 -0.814115 175 290 10.41 0.49 0.89 -0.187 -0.614 0.438 0.429 -0.625116 174 290 10.99 0.54 0.98 -1.824 -2.047 -0.705 0.220 -1.12116 175 291 10.89 0.56 1.02 -1.721 -1.975 -0.466 0.254 -1.255116 176 292 10.774 0.59 1.07 -1.745 -1.926 -0.121 0.181 -1.624116 177 293 10.68 0.56 1.02 -1.276 -1.492 0.024 0.214 -1.300117 176 293 11.18 0.53 0.97 -1.854 -2.156 -0.883 0.301 -0.971117 177 294 11.07 0.48 0.87 -1.745 -1.580 -0.601 -0.165 -1.144118 176 294 11.82 0.54 0.98 -3.161 -3.376 -2.089 0.215 -1.072 δ = 1 M M (cid:88) i ∆ i (17)Mean deviation is given by | δ | = 1 M M (cid:88) i | ∆ i | (18)FIG. 5. ∆ calc and ∆ deformed WS plotted vs. charge number Z where they are dubbed prox and WS respectively.We note that the statistical parameters shown in Table II forsemiempirical formulas were obtained for 69 SHN in the re-gion ≤ Z ≤ [21]; owing to the similar sample size andsame region for charge number, we conclude that it is reason-able to make a comparison between them and our model. Wenote that there was no available data for δ and ∆ max − ∆ min forthe semiempirical formulas. We observe that our model Proxperforms better than all models except ImSahu. All in all,these results help to consolidate the importance of diffusenessin half-lives calculations; it shows that using accurate diffuse-ness parameter is more important than taking deformation into account. Moreover, we note that WS takes two parameters β and β (deformation parameters), while Prox takes only one,namely a eff (If we use our new fitted formula then we take noparameters at all). In addition, it shows the reasonableness ofthe mapping a eff = ( a α + a WS ) / proposed earlier. We expectthat by incorporating deformation, more accurate preforma-tion factor formula and etc. into our model, we can get evenmore accurate results.TABLE II. Statistical comparison between our model Proxvs. deformed WS, deformed Coulomb model (WS) vs.ImSahu vs. Sahu vs. Royer10 vs. VS vs. SemFIS vs. UNIV.Data for statistical parameters for semiempirical formulas aretaken from Ref. [21]. Model (cid:112) δ δ | δ | ∆ max − ∆ min Prox 0.41 0.064 0.36 1.65WS 0.79 0.01 0.67 3.44ImSahu 0.362 0.287Sahu 0.709 0.58Royer10 0.523 0.429VS 0.623 0.508SemFIS 0.504 0.413UNIV 0.477 0.392
Motivated by these results, we predicted the half-lives of150 SHN between ≤ Z ≤ with the aid of our newsemiempirical formula for a eff . We will compare our resultswith seven of the current most powerful semiempirical formu-las for α half-lives: Viola-Seaborg (VS), Royer (R), modifiedBrown 1 (mB1), modified Brown 2 (mB2), ImSahu, SemFISand UNIV. Half-lives predictions made by these 7 semiempir-ical formulas which we shall compare our model against aretaken from Refs. [14, 21].Predictions of half-lives for 35 SHN in the region ≤ Z ≤ are shown in Table III. The comparison is made be-tween our model Prox, Viola-Seaborg (VS), modified Brown1 (mB1), Royer (R), and modified Brown 2 (mB2) and theaverage of the four semiempirical formulas [14]. The resultsof our model are pretty consistent with the predictions of thefour formulas.TABLE III. Our model Prox log T calc vs. Viola-Seaborg log T VS vs. modified Brown formula 1 log T mB1 vs. Royer log T R vs. modified Brown formula 2 log T mB2 vs. average of the four semiempirical formulas log T avg and error ∆ avg between ourmodel and the average. Q α values and half-lives for the semiempirical formulas are from Ref. [14]. Z N A Q α log T calc log T VS log T mB1 log T R log T mB2 log T avg ∆ avg
105 159 264 8.95 1.096 1.179 1.181 1.078 1.118 1.139 0.043106 161 267 9.12 0.790 0.641 0.622 0.666 0.685 0.654 -0.013106 167 273 8.20 3.493 3.239 3.247 3.367 3.439 3.323 -0.170108 153 261 10.97 -2.402 -3.164 -3.197 -3.228 -3.335 -3.231 -0.830108 164 272 9.60 -0.123 -0.558 -0.613 -0.597 -0.562 -0.583 -0.460108 169 277 8.85 1.910 1.859 1.822 1.88 1.928 1.872 -0.036110 165 275 10.38 -1.627 -1.483 -1.503 -1.590 -1.573 -1.537 0.090110 166 276 10.23 -1.310 -1.603 -1.645 -1.749 -1.714 -1.678 -0.369110 177 278 9.94 -0.651 -0.919 -0.970 -0.952 -0.907 -0.937 -0.287111 165 276 10.44 -1.434 -1.067 -1.072 -0.983 -1.001 -1.031 0.403112 160 272 11.20 -2.402 -3.271 -3.257 -3.490 -3.563 -3.395 -0.993112 161 273 11.06 -2.199 -2.543 -2.517 -2.565 -2.650 -2.569 -0.370112 162 274 10.92 -1.980 -2.703 -2.703 -2.838 -2.900 -2.786 -0.806112 163 275 10.79 -1.772 -1.961 -1.949 -1.980 -2.054 -1.986 -0.214112 164 276 10.65 -1.525 -2.107 -2.120 -2.149 -2.203 -2.145 -0.620112 165 278 10.37 -0.985 -1.478 -1.507 -1.421 -1.468 -1.469 -0.484112 166 279 10.23 -0.692 -0.704 -0.721 -0.709 -0.768 -0.726 -0.034113 167 280 10.69 -1.551 -1.192 -1.170 -1.085 -1.126 -1.143 0.416114 164 278 11.55 -2.990 -3.594 -3.529 -3.778 -3.814 -3.679 -0.691114 165 279 11.41 -2.780 -2.876 -2.801 -2.923 -2.961 -2.890 -0.111114 166 280 11.28 -2.577 -3.047 -3.000 -3.151 -3.180 -3.095 -0.518114 167 281 11.14 -2.339 -2.315 -2.260 -2.362 -2.394 -2.333 0.006114 168 282 11.00 -2.086 -2.472 -2.445 -2.491 -2.515 -2.481 -0.390114 169 283 10.87 -1.845 -1.726 -1.691 -1.770 -1.798 -1.746 0.098114 170 284 10.73 -1.563 -1.868 -1.862 -1.795 -1.816 -1.835 -0.272114 171 285 10.59 -1.267 -1.107 -1.093 -1.146 -1.171 -1.129 0.137115 171 286 11.04 -2.093 -1.566 -1.500 -1.447 -1.455 -1.492 0.600115 176 291 10.35 -0.549 -0.207 -0.207 -0.224 -0.216 -0.214 0.335116 171 287 11.50 -2.886 -2.664 -2.554 -2.731 -2.715 -2.666 0.220116 172 288 11.36 -2.628 -2.832 -2.753 -2.819 -2.808 -2.803 0.175116 173 289 11.22 -2.356 -2.097 -2.012 -2.164 -2.148 -2.105 0.250117 175 292 11.4 -2.607 -1.934 -1.812 -1.798 -1.768 -1.828 0.772118 175 293 11.85 -3.367 -3.010 -2.833 -3.089 -3.020 -2.988 0.378118 177 295 11.58 -2.847 -2.463 -2.316 -2.545 -2.479 -2.451 0.395
TABLE IV. Statistical comparison showing how consistent our model Prox is on one hand with VS, mB1, Royer, and mB2 onthe other.
Model (cid:112) δ δ | δ | ∆ max − ∆ min Prox 0.43 -0.09 0.34 1.77
The most relevant statistical parameters are shown in TableIV. We can see that the values of the parameters are consistentwith the ones in Table II reassuring us further to the utility andviability of our fit.Next, we predict the half-lives of 115 SHN in the ≤ Z ≤ region and compare our results to that of ImSahu,SemFIS and UNIV semiempirical formulas [21]. The predic-tions are compiled in Table V. To perform meaningful statis-tical analysis, we considered logarithm of half-lives values inwhich the three semiempirical formulas converge within 10 % ;hence, we define (cid:15) as [14], (cid:15) = max( | log T i | ) − min( | log T j | )max( | log T i | ) (19)where the indices i and j can be 1, 2, and 3 representingImSahu, SemFIS, and UNIV, respectively. After looking forSHN with (cid:15) < . , we took the average of the three formulas log T avg = (log T ImSahu + log T SemFIS + log T UNIV ) / thendefined ∆ avg = log T avg − log T calc to measure consistencyof our results with that of the other formulas. The result of thisanalysis is shown in Table VI. The rms error is 0.526 which isstill considered acceptable with current standards.TABLE V. Half-lives calculations for our model Prox log T calc vs. ImSahu log T ImSahu vs. SemFIS log T SemFIS vs. UNIV log T UNIV . Q α values and half-lives for the semiempirical formulas are from Ref. [21]. Z N A Q α log T calc log T ImSahu log T SemFIS log T UNIV
118 160 278 13.89 -5.693 -6.69 -7.32 -7.26118 161 279 13.78 -5.668 -8.25 -6.44 -6.4118 162 280 13.71 -5.705 -6.46 -6.90 -6.99118 163 281 13.76 -5.948 -8.14 -6.31 -6.40118 164 282 13.49 -5.606 -6.15 -6.42 -6.64118 165 283 13.33 -5.447 -7.32 -5.46 -5.68118 166 284 13.23 -5.389 -5.75 -5.87 -6.21118 167 285 13.07 -5.207 -6.77 -4.91 -5.24118 168 286 12.92 -5.029 -5.25 -5.2 -5.67118 169 287 12.8 -4.900 -6.19 -4.35 -4.76118 170 288 12.62 -4.636 -4.75 -4.62 -5.12118 171 289 12.59 -4.668 -5.71 -3.93 -4.39118 172 290 12.6 -4.775 -4.77 -4.59 -5.11118 173 291 12.42 -4.479 -5.30 -3.60 -4.08118 174 292 12.24 -4.167 -4.12 -3.87 -4.42118 175 293 12.24 -4.231 -4.85 -3.28 -3.74118 176 294 11.82 -3.355 -3.30 -3.02 -3.53118 177 295 11.9 -3.585 -4.06 -2.62 -3.05118 178 296 11.75 3.285 -3.21 -2.97 -3.43118 179 297 12.1 -4.106 -4.41 -3.19 -3.51118 180 298 12.18 -4.307 -4.19 -4.06 -4.39118 181 299 12.05 -4.041 -4.22 -3.23 -3.44118 182 300 11.96 -3.852 -3.79 -3.76 -3.95118 183 301 12.02 -3.990 -4.08 -3.36 -3.41118 184 302 12.04 -4.030 -4.03 -4.13 -4.16118 185 303 12.6 -5.213 -5.18 -4.77 -4.62118 186 304 13.12 -6.227 -6.20 -6.49 -6.32118 187 305 12.91 -5.792 -5.69 -5.59 -5.25118 188 306 12.48 -4.885 -5.05 -5.52 -5.12118 189 307 11.92 -3.631 -3.63 -3.90 -3.28118 190 308 12.2 -4.200 -2.34 -3.10 -2.35119 161 280 14.3 -6.224 -7.32 -6.51 -6.30119 162 281 14.16 -6.154 -7.26 -6.97 -6.96119 163 282 14 -6.038 -6.90 -5.90 -5.84119 164 283 13.76 -5.766 -6.60 -6.19 -6.32119 165 284 13.57 -5.566 -6.24 -5.05 -5.14119 166 285 13.61 -5.777 -6.34 -5.84 -6.10119 167 286 13.43 -5.575 -6.06 -4.73 -4.92119 168 287 13.28 -5.415 -5.76 -5.17 -5.54119 169 288 13.23 -5.435 -5.77 -4.31 -4.59119 170 289 13.16 -5.408 -5.54 -4.91 -5.35119 171 290 13.07 -5.334 -5.54 -3.98 -4.33119 172 291 13.05 -5.388 -5.02 -3.35 -3.74119 173 292 12.9 -5.176 -4.78 -4.20 -4.69119 174 293 12.72 -4.890 -4.59 -2.19 -3.28119 175 294 12.73 -4.891 -4.90 -3.33 -3.52119 185 304 12.93 -5.677 -5.71 -4.39 -4.28119 186 305 13.42 -6.609 -5.97 -6.15 -6.07119 187 306 13.2 -6.172 -6.27 -5.10 -4.82119 188 307 12.78 -5.316 -4.80 -5.18 -4.91119 189 308 12.06 -3.752 -4.11 -3.11 -2.59119 190 309 11.37 -2.097 -1.81 -2.49 -1.92120 163 283 14.31 -6.238 -9.02 -6.87 -6.83120 164 284 13.99 -5.843 -6.45 -6.91 -7.01
TABLE V. (
Continued ). Z N A Q α log T calc log T ImSahu log T SemFIS log T UNIV
120 165 285 13.89 -5.818 -8.25 -6.05 -6.17120 166 286 14.03 -6.211 -6.58 -6.88 -7.11120 167 287 13.85 -6.029 -8.11 -5.90 -6.13120 168 288 13.73 -5.942 -6.15 -6.28 -6.63120 169 289 13.71 -6.028 -7.79 -5.59 -5.92120 170 290 13.7 -6.124 -6.16 -6.18 -6.61120 171 291 13.51 -5.880 -7.36 -5.19 -5.60120 172 292 13.47 -5.905 -5.82 -5.74 -6.24120 173 293 13.4 -5.865 -7.09 -4.98 -5.43120 174 294 13.24 -5.644 -5.47 -5.31 -5.85120 175 295 13.27 -5.779 -6.77 -4.76 -5.23120 176 296 13.34 -5.983 -5.72 -5.54 -6.06120 177 297 13.14 -5.659 -6.46 -4.55 -5.02120 178 298 13.01 -5.457 -5.18 -4.97 -5.48120 179 299 13.26 -5.994 -6.60 -4.87 -5.27120 180 300 13.32 -6.149 -5.81 -5.66 -6.09120 181 301 13.06 -5.676 -6.15 -4.59 -4.93120 182 302 12.89 -5.361 -5.08 -4.95 -5.31120 183 303 12.81 -5.216 -5.60 -4.23 -4.48120 184 304 12.76 -5.123 -4.89 -4.84 -5.09120 185 305 13.28 -6.158 -6.40 -5.31 -5.40120 186 306 13.79 -7.102 -6.82 -6.95 -7.01120 187 307 13.52 -6.594 -6.74 -5.94 -5.86120 188 308 12.97 -5.512 -5.42 -5.65 -5.56120 189 309 12.16 -3.779 -4.05 -3.51 -3.25120 190 310 11.5 -2.211 -2.41 -2.82 -2.48121 165 286 14.34 -6.279 -7.31 -7.26 -5.96121 166 287 14.53 -6.752 -7.56 -7.53 -7.15121 167 288 14.46 -6.778 -7.56 -7.37 -6.18121 168 289 14.4 -6.813 -7.36 -7.23 -6.97121 169 290 14.42 -6.976 -7.56 -7.24 -6.15121 170 291 14.4 -7.064 -7.35 -7.18 -7.00121 171 292 14.31 -7.024 -7.45 -7.01 -6.00121 172 293 14.1 -6.766 -6.86 -6.63 -6.54121 173 294 14.1 -6.865 -7.17 -6.63 -5.69121 174 295 13.98 -6.744 -6.66 -6.42 -6.37121 175 296 14.01 -6.881 -7.08 -6.48 -5.57121 176 297 14.12 -7.152 -6.89 -6.68 -6.64121 177 298 13.89 -6.812 -6.94 -6.30 -5.39121 178 299 13.65 -6.435 -6.09 -5.88 -5.86121 179 300 13.81 -6.783 -6.87 -6.21 -5.29121 180 301 13.82 -6.847 -6.38 -6.27 -6.19121 181 302 13.49 -6.275 -6.38 -5.71 -4.75121 182 303 13.31 -5.962 -5.48 -5.42 -5.31121 183 304 13.28 -5.928 -6.06 -5.43 -4.40121 184 305 13.27 -5.925 -5.40 -5.48 -5.27121 185 306 13.81 -6.948 -7.06 -6.55 -5.38121 186 307 14.34 -7.883 -7.23 -7.54 -7.15121 187 308 14.07 -7.409 -7.55 -7.17 -5.85121 188 309 13.26 -5.896 -5.38 -5.81 -5.31121 189 310 12.46 -4.238 -4.66 -4.34 -2.89121 190 311 11.81 -2.747 -2.44 -3.06 -2.37
TABLE VI. Measure of consistency of our model Prox with that of ImSahu, SemFIS, and UNIV.
Model (cid:112) δ δ | δ | ∆ max − ∆ min Prox 0.526 -0.013 0.4217 2.46 VI. DISCUSSION AND CONCLUSIONS
The present analysis and results leave us with several con-clusions and remarks:• The effect of diffuseness can no longer be ignored byadopting it as a constant for all SHN; even very slightvariations as small as 0.1 fm can induce an error in half-life as large as 0.69; this is especially true for systemswhich we expect to have large diffuseness; the effectis especially pronounced when proximity potential isadopted in which we find non-linear (convex) variationof the logarithm of half-life with diffuseness; we con-clude that diffuseness is a great bottleneck acting as alimiting factor against accurate half-lives calculations.• Present calculations and comparisons show that tak-ing true values of diffuseness parameter into account ismore critical than deformation effects; this was shownwhen we compared our model with WS model.• The use of the mapping a eff = ( a α + a WS ) / producedresults for 68 SHN that are in great agreement withavailable experimental data; it outperformed all mod-els except the ImSahu model. This shows the viabilityand utility of this mapping, which can be adopted forfuture use.• Our predictions of α half-lives of 150 SHN using ournewly proposed fitted formula is in good agreementwith other semiempirical formulas; this is especially true for the region ≤ Z ≤ which was the re-gion of the original fit. By using a bigger sample sizeover a more extended region, a semiempirical formulabetter than the one at hand can be produced to be usedin half-lives calculations.• For our predictions, we did not incorporate centripetalcontribution V l ( r ) due to the lack of information aboutangular momentum of the systems under study. Incor-porating these data will improve the accuracy of the re-sults.• Present investigations and calculations show that a eff isbest represented as a linear function of charge and neu-tron numbers.• Present calculations corroborate the viability of theProx model; we expect that by taking further effects intoaccount including deformation, more accurate parity-dependent preformation factor and etc., the errors willbe reduced even further.• A potentially fruitful research direction is to investigatethe role of diffuseness in half-life using other modelsfor nuclear potential.• We stress that future research should invest an appre-ciable amount of energy in calculating and extractingaccurate diffuseness parameters values. [1] M. Ismail, W. M. Seif, A. Adel, and A. Abdurrahman, Nucl.Phys. A. , 202 (2017).[2] D. N. Basu, Phys. Lett. B , 90 (2003).[3] Z. Ren, C. Xu, and Z. Wang, Phys. Rev. C , 034304 (2004).[4] G. Royer and R. Moustabchir, Nucl. Phys. A , 182 (2001).[5] G. L. Zhang, H. B. Zheng, and W. W. Qu, Eur. Phys. J. A ,10 (2013).[6] Y. J. Yao, G. L. Zhang, W. W. Qu, and J. Q. Qian, Eur. Phys. J.A , 122 (2015).[7] H. F. Zhang, G. Royer, and J. Q. Li, Phys. Rev. C , 027303(2011).[8] V. Y. Denisov and H. Ikezoe, Phys. Rev. C , 064613 (2005).[9] C. Xu and Z. Ren, Nucl. Phys. A , 303 (2005).[10] C. Xu and Z. Ren, Phys. Rev. C , 014304 (2006).[11] X. D. Sun, J. G. Deng, D. Xiang, P. Guo, and X. H. Li, Phys.Rev. C , 044303 (2017).[12] V. Dehghani, S. A. Alavi, and K. Benam, Mod. Phys. Lett. A , 1850080 (2018).[13] C. L. Guo, G. L. Zhang, and M. Li, Int. J. Mod. Phys. E ,1550040 (2015).[14] A. I. Budaca, R. Budaca, and I. Silisteanu, Nucl. Phys. A ,60 (2016).[15] S. Guo, X. Bao, Y. Gao, J. Li, and H. Zhang, Nucl. Phys. A ,110 (2015).[16] L. Zheng, G. L. Zhang, J. C. Yang, and W. W. Qu, Nucl. Phys.A , 70 (2013).[17] J. Błocki, J. Randrup, W. J. ´Swia¸tecki, and C. F. Tsang, Ann.Phys. (NY) , 427 (1977).[18] W. D. Myers and W. J. ´Swia¸tecki, Phys. Rev. C , 044610(2000).[19] P. M¨oller and J. R. Nix, Nucl. Phys. A , 117 (1981).[20] R. K. Gupta, D. Singh, R. Kumar, and W. Greiner, J. Phys. G:Nucl. Part. Phys. , 075104 (2009).[21] S. Zhang, Y. Zhang, J. Cui, and Y. Wang, Phys. Rev. C95