1. Introduction
The synthesis of superheavy nuclei(SHN) has always been a hot topic in nuclear physics. Many studies have been done on the decay properties of superheavy nuclei, which provides valuable guidance for experiments. In the experiments, elements 107–112 have been synthesized in cold-fusion reactions [1, 2]. While elements 113–118 have been successfully synthesized in hot-fusion reactions [3–8]. Moreover, the syntheses of new elements 119 and 120 are also in progress [9, 10]. For SHN, α decay is one of the dominant decay modes, which can be used as a powerful tool to identify new elements or new isotopes by detecting the α decay chains [11–17]. Furthermore, the study on α decay can provide valuable nuclear structure information, such as ground-state energies, half-lives, nuclear spins, shell effects, nuclear deformation and so on [18–24].
α decay was first observed as an unknown radioactive phenomenon by Becquerel in 1896 and further explained as a process in which the parent nucleus emits a 4He particle by Rutherford and Geiger in 1908 [25]. Subsequently, it was successfully explained by Gamow [26] and by Condon and Gurney [27] as a quantum tunneling effect. Since then, many phenomenological and/or macroscopic-microscopic models have been proposed to study this process, such as the generalized liquid drop model [28, 29], the density-dependent M3Y effective interaction [30, 31], the cluster model [32, 33], the coupled channel approach [34, 35], the two-potential approach [36, 37], etc. Meanwhile, a lot of empirical formulae have also been proposed to calculate α-decay half-lives, which are in good agreement with experimental data [38–48]. More importantly, these works are often used to predict α-decay half-lives of unknown superheavy nuclei to aid in experimental design.
It is well known that α-decay half-life is extremely sensitive to the decay energy Qα. For instance, an uncertainty of 1 MeV in α-decay energy results in an uncertainty of α-decay half-life ranging from 103 to 105 times in the heavy and superheavy regions [49]. However, it is still a challenge to accurately predict the values of α-decay energy for superheavy nuclei, which brings great difficulties to design experiments for the synthesis of new elements 119 and 120. Meanwhile, in recent years, many experimental data have been accurately measured and/or added in the heavy and superheavy regions. Whether there is a more accurate formula to calculate α-decay energy with fewer parameters or not seems to be a worthwhile attempt. For this purpose, based on the liquid-drop model, we put forward an improved formula to evaluate α-decay energy for heavy and superheavy nuclei. The calculated results are in good agreement with the experimental data. The corresponding average and standard deviation of 209 nuclei are 0.141 and 0.190MeV, respectively. Moreover, we generalize this formula to predict the α-decay energies for nuclei with Z = 117, 118, 119 and 120. For comparison, the reliable formulae proposed by Dong J M et al (DZ formula) [50], Dong T K et al (DR formula) [51] and WS3+ nuclear mass model proposed by Wang N et al (WS3+) [52] are also used. The corresponding predictions of these formulae are basically consistent.
This article is organized as follows. In section 2 , the theoretical framework is briefly described. The detailed results and discussion are presented in section 3 . Finally, a succinct summary is given in section 4 .
2. Theoretical framework
Based on the liquid-drop model, the standard Bethe-Weizsäcker formula of nuclear binding energy is expressed as
$\begin{eqnarray}\begin{array}{rcl}B(Z,A) & = & {B}_{v}+{B}_{s}+{B}_{c}+{B}_{a}+{B}_{p}\\ & = & {a}_{v}A-{a}_{s}{A}^{2/3}-{a}_{c}{Z}^{2}{A}^{-1/3}\\ & & -{a}_{a}{\left(\displaystyle \frac{A}{2}-Z\right)}^{2}{A}^{-1}+{B}_{p},\end{array}\end{eqnarray}$
where Bv, Bs, Bc, Ba and Bp are the volume, surface, Coulomb, symmetry and pairing energy, respectively. A and Z are the mass and proton numbers of the nucleus, respectively.The α-decay energy is related to the parent and daughter nuclei by the relation2 ) and (3 ), we obtain the α-decay energy formula without considering shell correction. It can be given as
$\begin{eqnarray}{Q}_{\alpha }={B}_{\alpha }+B(Z-2,A-4)-B(Z,A)={B}_{\alpha }+{\rm{\Delta }}B,\end{eqnarray}$
where Bα, B(Z − 2, A − 4) and B(Z, A) are the binding energy of the α particle, daughter nucleus and parent nucleus, respectively. If the change in binding energy is smooth as Z and A, ΔB can be approximated as $\begin{eqnarray}{\rm{\Delta }}B\approx \displaystyle \frac{\partial B}{\partial Z}{\rm{\Delta }}Z+\displaystyle \frac{\partial B}{\partial A}{\rm{\Delta }}A,\end{eqnarray}$
with ΔZ = –2 and ΔA = –4. Combining equations ( $\begin{eqnarray}\begin{array}{rcl}{Q}_{\alpha } & = & {B}_{\alpha }+B(Z-2,A-4)-B(Z,A)\\ & = & {B}_{\alpha }-4{a}_{v}+\displaystyle \frac{8}{3}{a}_{s}{A}^{-1/3}+{\rm{\Delta }}{B}_{p}\\ & & +4{a}_{c}\displaystyle \frac{Z}{{A}^{1/3}}\left(1-\displaystyle \frac{Z}{3A}\right)-{a}_{a}{\left(\displaystyle \frac{N-Z}{A}\right)}^{2}\\ & = & {a}_{1}+{a}_{2}{A}^{-1/3}+{\rm{\Delta }}{B}_{p}\\ & & +{a}_{3}\displaystyle \frac{Z}{{A}^{1/3}}\left(1-\displaystyle \frac{Z}{3A}\right)+{a}_{4}{\left(\displaystyle \frac{N-Z}{A}\right)}^{2}.\end{array}\end{eqnarray}$
In the above equation, the second term i.e. a2A−1/3 denoted the contribution of surface energy to α-decay energy can be incorporated into the constant term since this term approaches a small constant with the mass number A of the parent nucleus increasing or decreasing, especially for SHN. Meanwhile, the third term i.e. ΔBp denoted the contribution of the paring energy to α-decay energy, can be ignored since the number of nucleons for the parent and daughter nuclei are even or odd at the same time in the α decay process. Consequently, the symmetry energy and Coulomb energy became the main contributions to α-decay energy. After the above discussion, the α-decay energy Qα can be written as $\begin{eqnarray}{Q}_{\alpha }={a}_{1}+{a}_{2}\displaystyle \frac{Z}{{A}^{1/3}}\left(1-\displaystyle \frac{Z}{3A}\right)+{a}_{3}{\left(\displaystyle \frac{N-Z}{A}\right)}^{2},\end{eqnarray}$
which is basically consistent with the DZ formula for α-decay energies of heavy and superheavy nuclei without considering shell correction [50].It is well-known that the standard Bethe-Weizsäcker formula completely depends on the liquid-drop model, which only reflects the average trend of energy change. However, when the nucleon number is close to the region near the shell, both binding energy and α-decay energy change abruptly [53, 54]. This difference can lead to large deviations between experimental data and theoretical results. Therefore, it is necessary to quantify the deviations caused by the shell effect and introduce shell correction. However, it is a tricky issue to deal with the shell effect. In 1960, Strutinsky [55] treated the shell effect as the deviation between the nonuniform level density and a uniform one and further added this deviation to the liquid-drop model. Based on this method, the macroscopic-microscopic models [49, 56] are developed to study the properties of superheavy nuclei and achieve great success, which proves the shell effect can be considered separately. Subsequently, Ren et al. use analytical formulae to consider shell correction and further calculate the binding energy and α- decay energy [51, 53, 54], which also obtain satisfactory results. Using the same method, we systematically calculate 209 α-decay energies and plot the deviations between the experimental data and calculated ones by using equation (5 ) in figure 1. From this figure, especially in regions I and II, one can see that the deviations have an obvious symmetry with different amplitudes similar to the sine function, which is largely due to the shell effect. Whether similar analytical formulae can be used to better express shell correction for different regions attracts our interest. After many attempts, the shell correction is expressed by the first derivative of the normalized Gaussian function, which can be written as
$\begin{eqnarray}f(t)=\displaystyle \frac{1}{\sqrt{2\pi }}t\exp \left[-\displaystyle \frac{1}{2}{t}^{2}\right].\end{eqnarray}$
Figure 1. The deviations between experimental α-decay energies and calculated ones by using equation ( |
Combining equations (5 ) and (6 ), we obtain the final formula of α-decay energy. It can be expressed as
$\begin{eqnarray}\begin{array}{rcl}{Q}_{\alpha } & = & {a}_{1}+{a}_{2}\displaystyle \frac{Z}{{A}^{1/3}}\left(1-\displaystyle \frac{Z}{3A}\right)+{a}_{3}{\left(\displaystyle \frac{N-Z}{A}\right)}^{2}\\ & & +{a}_{4}\left(\displaystyle \frac{N-{N}_{0}}{{a}_{5}}\right)\exp \left[-\displaystyle \frac{1}{2}{\left(\displaystyle \frac{N-{N}_{0}}{{a}_{5}}\right)}^{2}\right]\\ & & +{a}_{6}\left(\displaystyle \frac{N-{N}_{1}}{{a}_{7}}\right)\exp \left[-\displaystyle \frac{1}{2}{\left(\displaystyle \frac{N-{N}_{1}}{{a}_{7}}\right)}^{2}\right].\end{array}\end{eqnarray}$
In this formula, N0 and N1 represent the center point corresponding to the correction function in regions I and II, respectively. The position of N0 and N1, properties of the correction function and detailed calculations will be shown in the next section.3. Results and discussion
Based on the liquid-drop model, considering the shell correction, we put forward an improved formula to evaluate α-decay energy for heavy and superheavy nuclei. In this formula, the shell correction is described by the first derivative of the normalized Gaussian function. In order to exhibit more details of this function, the shape of this function is plotted in figure 2. One can see that the shape of this function is similar to the deviations between experimental data and calculated ones in figure 1. Meanwhile, this function is also localized, which does not have an influence on other nuclei. Based on the properties of this function, it is critical to find the center of this function, that is, the position of the neutron number corresponding to t = 0 in these two regions. Based on equation (7 ), we use the least square method to fit 209 experimental α decay energies and obtain N0 = 154 and N1 = 170 as the matched center point corresponding to this correction function in the region I and II, respectively. Meanwhile, the corresponding values of parameters are as follows $\left\{\begin{array}{l}{a}_{1}=-26.3930\\ {a}_{2}=2.7500\\ {a}_{3}=-95.1273\\ {a}_{4}=0.3753\\ {a}_{5}=2.5528\\ {a}_{6}=-1.0495\\ {a}_{7}=3.7382.\end{array}\right.$
Figure 2. The shape of the first derivative of the normalized Gaussian function. |
For these parameters, a5 and a7 are dimensionless, and other parameters are in MeV. The a2 and a3 are the coefficients of Coulomb energy and symmetry energy, respectively. Where a4 and a6 are the amplitudes of the correction function, their values depend on the extent of deviations caused by the shell effects. From figure 1, it can be seen that the wave crest and valley of the deviations in region II are exactly opposite to the correction function, which is the reason why a6 is negative.
For a more intuitive comparison of the results with or without introducing shell correction, the deviations between experimental α-decay energy and calculated ones by using equations (5 ) and (7 ) are plotted in figure 3. From this figure, one can see that there is a marked improvement by introducing shell correction. Noticeably, the shell effects are complex for nuclei with N ≥ 170 and the experimental data can hardly be reproduced. While the calculated results by using equation (7 ) can reproduce the experimental α-decay energy within 0.1 MeV for most nuclei in this region. More importantly, it is well known that the shell effects will be strong and the deviations between experimental α-decay energies and predicted ones will be very large when the experimental synthesis approaches the next neutron magic number. This region is still far from the neutron magic number N = 184 predicted by the macroscopic-microscopic model and Skyrme-Hartree–Fock model [49, 56, 57]. Therefore, it is feasible to give a reliable prediction for nuclei with 170 ≤ N ≤ 180.
Figure 3. The deviations between experimental α-decay energies and calculated ones for 209 nuclei. (a) for calculations by using equation ( |
The detailed calculations of 209 α-decay energies are shown in table 1. In this table, the first two columns denote the parent nuclei of α decay and corresponding values of decay energy Qα, respectively. The last four columns denote the calculated results by using equation (7 ), the DR formula, the DZ formula, and WS3+, respectively. From this table, it can be found that the value of α-decay energy in the isotope chains decreases gradually with the increase of the number of neutrons, which is consistent with the conclusion of the quantum tunneling effect. The lower the decay energy, the more difficult it is for the α particle to penetrate the barrier from the parent nuclei and therefore the more stable nucleus they are, which also indicates that superheavy nuclei should be neutron-rich [50]. Meanwhile, from the results calculated by using equation (7 ), it can be seen that the calculated results can reproduce the experimental α-decay energies within 0.3 MeV for most nuclei. After careful analysis of the nuclei with large deviations, we divide them into two categories. The first category mainly includes 255Md, 255No, 256No, 257No, 257Lr and 258Lr. These nuclei are mainly concentrated in the region near the proton number Z = 102 and the neutron number N = 154, which we believe is determined by the complex shell effects around Z = 100 and N = 152. It should be pointed out that this situation also occurs in the calculated results of the binding energy for heavy and superheavy nuclei [53, 54]. The second category mainly includes 270Hs, 267Ds, 269Ds and 273Ds, the 270Hs has been predicted to be a deformed doubly magic nucleus by the macroscopic-microscopic models [49, 56] and this is supported by experiments [58, 59]. For these nuclei, their proton numbers are near Z = 108 or the neutron numbers are around N = 162. These factors may result in large deviations. Since the shell effect is a complex mechanism, there is no ideal approach to deal with this effect at present. In this work, the shell effect is considered by introducing analytic formula, which is an approximate method. Following this approach, the shell effect can not be completely described by analytical formulae bur what can be done is to search for a more appropriate analytical formula to minimize this approximate difference as much as possible. After the above discussion, it is reasonable to have a few nuclei with large deviations among 209 nuclei.
The $\overline{\sigma }$ and $\sqrt{{\sigma }^{2}}$ represent the average and standard deviation between experimental data and calculated ones. In this work, they are defined as follows8 ) and (9 ), we calculate the average and standard deviation between experimental data and calculated ones by using the DZ formula, DR formula, WS3+, and equation (7 ), respectively. The detailed calculated results are listed in table 2. From this table, it can be seen that our formula achieves satisfactory results with only seven parameters. In addition, the differences between experimental data and calculated ones for 209 nuclei by using these formulae and equation (7 ) are plotted in figure 4. One can see that compared with the DZ formula and DR formula, our improved 7-parameter formula can reproduce experimental data better.
$\begin{eqnarray}\overline{\sigma }=\displaystyle \sum _{{i}=1}^{209}| {Q}_{\alpha }^{{i}}\exp -{Q}_{\alpha }^{{i}}\mathrm{cal}| /209,\end{eqnarray}$
$\begin{eqnarray}\sqrt{{\sigma }^{2}}={\left[\displaystyle \sum _{{i}=1}^{209}{\left({Q}_{\alpha }^{{i}}\exp -{Q}_{\alpha }^{{i}}\mathrm{cal}\right)}^{2}/209\right]}^{\tfrac{1}{2}}.\end{eqnarray}$
Using equations (
Figure 4. The deviations between experimental α-decay energies and calculated ones for 209 nuclei. The black square, yellow hexagon, red ball, and green triangle denote the deviations calculated by the DZ formula, DR formula, equation ( |
Table 1. Calculations of α-decay energies (in MeV) for 209 nuclei. ${Q}_{\alpha }^{\mathrm{cal}}$, ${Q}_{\alpha }^{\mathrm{DZ}}$, ${Q}_{\alpha }^{\mathrm{DR}}$ and ${Q}_{\alpha }^{\mathrm{WS}3+}$ denote the calculated ones by using equation ( |
| Nuclei | ${Q}_{\alpha }^{\exp }$ | ${Q}_{\alpha }^{\mathrm{cal}}$ | ${Q}_{\alpha }^{\mathrm{DZ}}$ | ${Q}_{\alpha }^{\mathrm{DR}}$ | ${Q}_{\alpha }^{\mathrm{WS}3+}$ | Nuclei | ${Q}_{\alpha }^{\exp }$ | ${Q}_{\alpha }^{\mathrm{cal}}$ | ${Q}_{\alpha }^{\mathrm{DZ}}$ | ${Q}_{\alpha }^{\mathrm{DR}}$ | ${Q}_{\alpha }^{\mathrm{WS}3+}$ |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 230Th | 4.770 | 4.238 | 3.771 | 4.374 | 4.786 | 236Np | 5.010 | 5.286 | 4.848 | 5.259 | 5.104 |
| 231Th | 4.213 | 4.069 | 3.595 | 4.162 | 4.342 | 237Np | 4.957 | 5.123 | 4.678 | 5.037 | 5.003 |
| 232Th | 4.082 | 3.899 | 3.418 | 3.941 | 3.969 | 238Np | 4.691 | 4.957 | 4.508 | 4.820 | 4.838 |
| 233Th | 3.745 | 3.727 | 3.240 | 3.712 | 3.799 | 239Np | 4.597 | 4.786 | 4.337 | 4.617 | 4.563 |
| 234Th | 3.672 | 3.555 | 3.062 | 3.481 | 3.699 | 240Np | 4.557 | 4.605 | 4.165 | 4.436 | 4.569 |
| 235Th | 3.376 | 3.381 | 2.882 | 3.255 | 3.457 | 241Np | 4.360 | 4.407 | 3.992 | 4.283 | 4.251 |
| 233Th | 3.333 | 3.201 | 2.702 | 3.043 | 3.133 | 242Np | 4.100 | 4.188 | 3.818 | 4.160 | 4.231 |
| 237Th | 3.196 | 3.012 | 2.521 | 2.853 | 3.103 | 234Pu | 6.310 | 6.263 | 5.859 | 6.383 | 6.368 |
| 231Pa | 5.150 | 4.756 | 4.306 | 4.889 | 5.071 | 235Pu | 5.951 | 6.106 | 5.696 | 6.184 | 6.030 |
| 232Pa | 4.627 | 4.590 | 4.133 | 4.680 | 4.665 | 236Pu | 5.867 | 5.948 | 5.532 | 5.975 | 5.755 |
| 233Pa | 4.375 | 4.423 | 3.959 | 4.461 | 4.273 | 237Pu | 5.748 | 5.789 | 5.367 | 5.759 | 5.773 |
| 234Pa | 4.076 | 4.255 | 3.785 | 4.236 | 4.178 | 238Pu | 5.593 | 5.629 | 5.201 | 5.540 | 5.514 |
| 235Pa | 4.101 | 4.086 | 3.609 | 4.008 | 4.062 | 239Pu | 5.245 | 5.466 | 5.033 | 5.326 | 5.318 |
| 236Pa | 3.755 | 3.914 | 3.433 | 3.785 | 3.835 | 240Pu | 5.256 | 5.298 | 4.865 | 5.125 | 5.138 |
| 237Pa | 3.795 | 3.738 | 3.255 | 3.575 | 3.685 | 241Pu | 5.140 | 5.121 | 4.696 | 4.947 | 5.135 |
| 238Pa | 3.628 | 3.551 | 3.077 | 3.389 | 3.381 | 242Pu | 4.984 | 4.925 | 4.526 | 4.798 | 4.954 |
| 232U | 5.414 | 5.267 | 4.832 | 5.395 | 5.340 | 243Pu | 4.757 | 4.709 | 4.355 | 4.678 | 4.850 |
| 233U | 4.909 | 5.104 | 4.663 | 5.189 | 4.965 | 244Pu | 4.666 | 4.480 | 4.184 | 4.581 | 4.640 |
| 234U | 4.858 | 4.940 | 4.492 | 4.974 | 4.661 | 235Am | 6.576 | 6.749 | 6.361 | 6.866 | 6.824 |
| 235U | 4.678 | 4.775 | 4.320 | 4.752 | 4.629 | 236Am | 6.260 | 6.596 | 6.201 | 6.669 | 6.506 |
| 236U | 4.573 | 4.608 | 4.148 | 4.527 | 4.476 | 238Am | 6.040 | 6.285 | 5.878 | 6.251 | 6.049 |
| 237U | 4.234 | 4.440 | 3.975 | 4.307 | 4.239 | 239Am | 5.922 | 6.128 | 5.715 | 6.035 | 6.036 |
| 238U | 4.270 | 4.266 | 3.800 | 4.100 | 4.022 | 240Am | 5.710 | 5.968 | 5.551 | 5.824 | 5.800 |
| 239U | 4.130 | 4.082 | 3.625 | 3.916 | 3.906 | 241Am | 5.638 | 5.803 | 5.385 | 5.626 | 5.692 |
| 240U | 4.035 | 3.882 | 3.449 | 3.761 | 3.811 | 242Am | 5.588 | 5.628 | 5.219 | 5.451 | 5.536 |
| 233Np | 5.630 | 5.769 | 5.350 | 5.893 | 5.696 | 243Am | 5.439 | 5.436 | 5.052 | 5.305 | 5.471 |
| 234Np | 5.356 | 5.609 | 5.184 | 5.690 | 5.481 | 244Am | 5.138 | 5.222 | 4.884 | 5.187 | 5.168 |
| 235Np | 5.194 | 5.448 | 5.016 | 5.478 | 5.185 | 245Am | 5.160 | 4.995 | 4.716 | 5.093 | 5.137 |
| 236Cm | 7.067 | 7.228 | 6.854 | 7.341 | 7.167 | 252Cf | 6.217 | 6.046 | 5.615 | 6.198 | 6.175 |
| 237Cm | 6.770 | 7.077 | 6.698 | 7.148 | 6.918 | 253Cf | 6.126 | 6.025 | 5.451 | 6.039 | 6.116 |
| 238Cm | 6.670 | 6.926 | 6.540 | 6.945 | 6.589 | 254Cf | 5.927 | 5.949 | 5.286 | 5.860 | 5.952 |
| 239Cm | 6.540 | 6.773 | 6.381 | 6.735 | 6.502 | 240Es | 8.260 | 8.479 | 8.142 | 8.538 | 8.340 |
| 240Cm | 6.398 | 6.619 | 6.221 | 6.522 | 6.493 | 241Es | 8.259 | 8.337 | 7.994 | 8.345 | 7.992 |
| 241Cm | 6.185 | 6.462 | 6.060 | 6.314 | 6.192 | 242Es | 8.160 | 8.193 | 7.844 | 8.144 | 8.006 |
| 242Cm | 6.216 | 6.300 | 5.898 | 6.120 | 6.260 | 243Es | 8.072 | 8.047 | 7.693 | 7.941 | 8.201 |
| 243Cm | 6.169 | 6.128 | 5.735 | 5.947 | 6.128 | 245Es | 7.909 | 7.746 | 7.389 | 7.556 | 7.883 |
| 244Cm | 5.902 | 5.938 | 5.571 | 5.804 | 5.840 | 247Es | 7.464 | 7.402 | 7.079 | 7.257 | 7.490 |
| 245Cm | 5.624 | 5.727 | 5.405 | 5.689 | 5.688 | 251Es | 6.597 | 6.634 | 6.448 | 6.918 | 6.685 |
| 246Cm | 5.475 | 5.503 | 5.240 | 5.598 | 5.437 | 252Es | 6.738 | 6.561 | 6.288 | 6.820 | 6.640 |
| 247Cm | 5.354 | 5.294 | 5.073 | 5.517 | 5.302 | 253Es | 6.739 | 6.543 | 6.127 | 6.692 | 6.753 |
| 248Cm | 5.162 | 5.138 | 4.905 | 5.431 | 5.191 | 254Es | 6.617 | 6.524 | 5.966 | 6.535 | 6.670 |
| 249Cm | 5.148 | 5.057 | 4.737 | 5.325 | 5.186 | 255Es | 6.436 | 6.450 | 5.803 | 6.359 | 6.484 |
| 250Cm | 5.170 | 5.031 | 4.568 | 5.189 | 5.103 | 243Fm | 8.690 | 8.652 | 8.317 | 8.600 | 8.512 |
| 243Bk | 6.874 | 6.789 | 6.402 | 6.606 | 6.782 | 246Fm | 8.379 | 8.214 | 7.871 | 8.021 | 8.411 |
| 244Bk | 6.779 | 6.620 | 6.242 | 6.436 | 6.663 | 247Fm | 8.258 | 8.053 | 7.720 | 7.860 | 8.222 |
| 245Bk | 6.455 | 6.433 | 6.081 | 6.296 | 6.392 | 248Fm | 7.995 | 7.875 | 7.567 | 7.728 | 8.008 |
| 246Bk | 6.070 | 6.225 | 5.919 | 6.184 | 6.111 | 249Fm | 7.709 | 7.675 | 7.414 | 7.625 | 7.826 |
| 247Bk | 5.890 | 6.004 | 5.756 | 6.095 | 5.916 | 250Fm | 7.557 | 7.463 | 7.260 | 7.545 | 7.580 |
| 248Bk | 5.830 | 5.798 | 5.592 | 6.017 | 5.803 | 251Fm | 7.425 | 7.264 | 7.104 | 7.475 | 7.313 |
| 249Bk | 5.521 | 5.644 | 5.427 | 5.934 | 5.656 | 252Fm | 7.154 | 7.119 | 6.948 | 7.400 | 7.195 |
| 239Cf | 7.760 | 8.019 | 7.668 | 8.082 | 7.862 | 253Fm | 7.198 | 7.048 | 6.791 | 7.304 | 7.183 |
| 240Cf | 7.711 | 7.874 | 7.516 | 7.886 | 7.592 | 254Fm | 7.308 | 7.032 | 6.632 | 7.179 | 7.320 |
| 242Cf | 7.517 | 7.578 | 7.210 | 7.475 | 7.619 | 255Fm | 7.240 | 7.017 | 6.473 | 7.025 | 7.139 |
| 244Cf | 7.329 | 7.271 | 6.899 | 7.084 | 7.446 | 256Fm | 7.025 | 6.945 | 6.314 | 6.851 | 7.050 |
| 245Cf | 7.258 | 7.105 | 6.742 | 6.918 | 7.279 | 257Fm | 6.864 | 6.801 | 6.153 | 6.674 | 6.751 |
| 246Cf | 6.862 | 6.921 | 6.584 | 6.780 | 6.950 | 244Md | 8.950 | 9.104 | 8.783 | 9.049 | 9.180 |
| 247Cf | 6.503 | 6.716 | 6.425 | 6.671 | 6.713 | 246Md | 8.890 | 8.822 | 8.493 | 8.658 | 9.237 |
| 248Cf | 6.361 | 6.497 | 6.265 | 6.586 | 6.529 | 247Md | 8.764 | 8.675 | 8.346 | 8.478 | 9.066 |
| 249Cf | 6.293 | 6.294 | 6.103 | 6.510 | 6.293 | 248Md | 8.500 | 8.517 | 8.197 | 8.321 | 8.840 |
| 250Cf | 6.128 | 6.143 | 5.942 | 6.430 | 6.217 | 250Md | 8.155 | 8.145 | 7.898 | 8.091 | 8.391 |
| 251Cf | 6.177 | 6.067 | 5.779 | 6.329 | 6.196 | 251Md | 7.963 | 7.935 | 7.746 | 8.014 | 8.161 |
| 253Md | 7.573 | 7.597 | 7.440 | 7.875 | 7.803 | 269Sg | 8.580 | 8.904 | 8.196 | 8.938 | 8.378 |
| 255Md | 7.906 | 7.515 | 7.130 | 7.659 | 7.886 | 260Bh | 10.400 | 10.272 | 10.112 | 10.509 | 10.328 |
| 257Md | 7.557 | 7.433 | 6.817 | 7.336 | 7.566 | 261Bh | 10.500 | 10.274 | 9.973 | 10.402 | 10.279 |
| 258Md | 7.271 | 7.291 | 6.659 | 7.162 | 7.223 | 262Bh | 10.319 | 10.277 | 9.833 | 10.266 | 10.261 |
| 251No | 8.752 | 8.608 | 8.374 | 8.551 | 8.929 | 270Bh | 9.060 | 9.370 | 8.677 | 9.402 | 8.680 |
| 252No | 8.549 | 8.400 | 8.226 | 8.476 | 8.611 | 271Bh | 9.420 | 9.350 | 8.528 | 9.299 | 9.061 |
| 253No | 8.415 | 8.207 | 8.076 | 8.412 | 8.445 | 272Bh | 9.300 | 9.315 | 8.379 | 9.151 | 9.106 |
| 254No | 8.226 | 8.067 | 7.925 | 8.343 | 8.332 | 274Bh | 8.940 | 9.062 | 8.079 | 8.711 | 8.780 |
| 255No | 8.428 | 8.002 | 7.774 | 8.252 | 8.422 | 263Hs | 10.730 | 10.717 | 10.287 | 10.705 | 10.966 |
| 256No | 8.582 | 7.991 | 7.621 | 8.133 | 8.610 | 264Hs | 10.591 | 10.666 | 10.148 | 10.551 | 10.626 |
| 257No | 8.477 | 7.981 | 7.467 | 7.983 | 8.375 | 265Hs | 10.470 | 10.542 | 10.009 | 10.395 | 10.561 |
| 259No | 7.854 | 7.775 | 7.158 | 7.643 | 7.784 | 266Hs | 10.346 | 10.369 | 9.868 | 10.252 | 10.326 |
| 252Lr | 9.164 | 9.064 | 8.844 | 9.004 | 9.165 | 267Hs | 10.038 | 10.189 | 9.726 | 10.135 | 10.026 |
| 253Lr | 8.918 | 8.859 | 8.698 | 8.932 | 8.870 | 270Hs | 9.070 | 9.863 | 9.296 | 9.929 | 9.184 |
| 254Lr | 8.822 | 8.669 | 8.552 | 8.871 | 8.672 | 273Hs | 9.650 | 9.780 | 8.858 | 9.614 | 9.614 |
| 255Lr | 8.556 | 8.532 | 8.404 | 8.804 | 8.664 | 275Hs | 9.450 | 9.531 | 8.562 | 9.178 | 9.297 |
| 256Lr | 8.850 | 8.469 | 8.255 | 8.716 | 8.777 | 266Mt | 10.996 | 10.981 | 10.461 | 10.832 | 11.216 |
| 257Lr | 9.070 | 8.461 | 8.105 | 8.599 | 8.850 | 275Mt | 10.480 | 10.158 | 9.186 | 9.878 | 10.246 |
| 258Lr | 8.904 | 8.453 | 7.954 | 8.453 | 8.765 | 276Mt | 10.100 | 9.994 | 9.040 | 9.639 | 9.962 |
| 255Rf | 9.055 | 9.124 | 9.020 | 9.323 | 8.941 | 278Mt | 9.580 | 9.372 | 8.745 | 9.076 | 9.495 |
| 256Rf | 8.926 | 8.989 | 8.875 | 9.259 | 8.960 | 267Ds | 11.780 | 11.415 | 10.908 | 11.263 | 11.765 |
| 257Rf | 9.083 | 8.929 | 8.729 | 9.174 | 9.094 | 269Ds | 11.510 | 11.072 | 10.636 | 11.014 | 11.316 |
| 258Rf | 9.196 | 8.924 | 8.582 | 9.059 | 9.241 | 270Ds | 11.117 | 10.926 | 10.499 | 10.933 | 10.880 |
| 261Rf | 8.650 | 8.723 | 8.134 | 8.585 | 8.632 | 271Ds | 10.870 | 10.822 | 10.360 | 10.874 | 10.720 |
| 256Db | 9.340 | 9.572 | 9.482 | 9.769 | 9.312 | 273Ds | 11.370 | 10.732 | 10.081 | 10.758 | 10.726 |
| 257Db | 9.206 | 9.440 | 9.340 | 9.707 | 9.338 | 272Rg | 11.197 | 11.259 | 10.811 | 11.310 | 11.182 |
| 258Db | 9.437 | 9.383 | 9.197 | 9.625 | 9.440 | 274Rg | 11.480 | 11.175 | 10.536 | 11.198 | 11.289 |
| 259Db | 9.620 | 9.380 | 9.052 | 9.513 | 9.451 | 278Rg | 10.850 | 10.903 | 9.976 | 10.544 | 11.096 |
| 259Sg | 9.765 | 9.830 | 9.658 | 10.070 | 9.751 | 279Rg | 10.530 | 10.643 | 9.834 | 10.276 | 10.663 |
| 260Sg | 9.901 | 9.830 | 9.516 | 9.961 | 9.940 | 280Rg | 10.149 | 10.290 | 9.691 | 9.990 | 10.193 |
| 261Sg | 9.714 | 9.830 | 9.373 | 9.822 | 9.651 | 277$\mathrm{Cn}$ | 11.620 | 11.579 | 10.713 | 11.406 | 11.953 |
| 262Sg | 9.600 | 9.774 | 9.229 | 9.664 | 9.650 | 281$\mathrm{Cn}$ | 10.430 | 10.740 | 10.155 | 10.439 | 10.493 |
| 263Sg | 9.400 | 9.645 | 9.084 | 9.502 | 9.180 | 278Nh | 11.990 | 12.015 | 11.161 | 11.840 | 12.287 |
| 282Nh | 10.780 | 11.184 | 10.613 | 10.882 | 10.961 | 288Mc | 10.650 | 10.653 | 10.969 | 10.756 | 10.227 |
| 284Nh | 10.280 | 10.374 | 10.334 | 10.329 | 10.124 | 289Mc | 10.490 | 10.497 | 10.831 | 10.577 | 10.096 |
| 285Nh | 10.010 | 10.023 | 10.193 | 10.084 | 9.787 | 290Mc | 10.410 | 10.424 | 10.692 | 10.427 | 10.093 |
| 286Nh | 9.790 | 9.763 | 10.052 | 9.870 | 9.436 | 290Lv | 11.000 | 10.936 | 11.283 | 11.015 | 10.878 |
| 285FI | 10.560 | 10.818 | 10.791 | 10.771 | 10.323 | 291Lv | 10.890 | 10.865 | 11.147 | 10.866 | 10.885 |
| 286FI | 10.360 | 10.469 | 10.652 | 10.528 | 9.944 | 292Lv | 10.791 | 10.842 | 11.010 | 10.740 | 10.917 |
| 287FI | 10.170 | 10.211 | 10.513 | 10.316 | 9.626 | 293Lv | 10.680 | 10.834 | 10.872 | 10.629 | 10.563 |
| 288FI | 10.076 | 10.053 | 10.373 | 10.135 | 9.472 | 293117 | 11.320 | 11.280 | 11.461 | 11.176 | 11.370 |
| 289FI | 9.950 | 9.978 | 10.233 | 9.982 | 9.427 | 294117 | 11.180 | 11.274 | 11.325 | 11.067 | 11.157 |
| 290FI | 9.860 | 9.951 | 10.091 | 9.851 | 9.361 | 294118 | 11.870 | 11.713 | 11.906 | 11.608 | 11.974 |
| 287Mc | 10.760 | 10.909 | 11.106 | 10.967 | 10.373 |
Table 2. The standard and average deviation (in MeV) between experimental α-decay energies and calculated ones denote as $\sqrt{{\sigma }^{2}}$ and $\overline{\sigma }$, respectively. * denotes the number of parameters is uncertain, the current one comes from the WS3 model [60]. |
| Cases | $\sqrt{{\sigma }_{\mathrm{DZ}}^{2}}$ | $\sqrt{{\sigma }_{\mathrm{DR}}^{2}}$ | $\sqrt{{\sigma }_{\mathrm{Ours}}^{2}}$ | $\sqrt{{\sigma }_{\mathrm{WS}3+}^{2}}$ |
| 209 | 0.482 | 0.223 | 0.190 | 0.167 |
| | ||||
| Cases | ${\overline{\sigma }}_{\mathrm{DZ}}$ | ${\overline{\sigma }}_{\mathrm{DR}}$ | ${\overline{\sigma }}_{\mathrm{Ours}}$ | ${\overline{\sigma }}_{\mathrm{WS}3+}$ |
| 209 | 0.416 | 0.177 | 0.141 | 0.121 |
| | ||||
| Parameters | 5 | 8 | 7 | 16* |
Finally, as an application, we extend this formula to predict the α-decay energies for nuclei with Z = 117, 118, 119 and 120. For comparison, the reliable DZ formula, DR formula, and WS3+ are also used. The corresponding predictions are listed in table 3. In this table, the first two columns denote the proton and mass number of the parent nucleus, respectively. The last four columns denote the α-decay energies predicted by using equation (7 ), DR formula, DZ formula, and WS3+, respectively. From this table, it can be seen that the predicted results of these formulae are basically consistent, which can provide useful information for experiments of synthesising new elements and isotopes.
Table 3. Predictions of α-decay energies (in MeV) for nuclei with Z = 117, 118, 119, and 120. ${Q}_{\alpha }^{\mathrm{cal}}$, ${Q}_{\alpha }^{\mathrm{DZ}}$, ${Q}_{\alpha }^{\mathrm{DR}}$ and ${Q}_{\alpha }^{\mathrm{WS}3+}$ denote the corresponding predictions by using equation ( |
| Z | A | ${Q}_{\alpha }^{\mathrm{cal}}$ | ${Q}_{\alpha }^{\mathrm{DZ}}$ | ${Q}_{\alpha }^{\mathrm{DR}}$ | ${Q}_{\alpha }^{\mathrm{WS}3+}$ | Z | A | ${Q}_{\alpha }^{\mathrm{cal}}$ | ${Q}_{\alpha }^{\mathrm{DZ}}$ | ${Q}_{\alpha }^{\mathrm{DR}}$ | ${Q}_{\alpha }^{\mathrm{WS}3+}$ |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 117 | 289 | 11.773 | 11.996 | 11.828 | 11.790 | 119 | 295 | 12.140 | 12.347 | 12.034 | 12.570 |
| 117 | 291 | 11.370 | 11.730 | 11.447 | 11.460 | 119 | 297 | 12.123 | 12.083 | 11.833 | 12.300 |
| 117 | 295 | 11.255 | 11.188 | 10.967 | 11.120 | 119 | 299 | 12.016 | 11.816 | 11.643 | 12.730 |
| 117 | 297 | 11.140 | 10.914 | 10.769 | 11.540 | 119 | 301 | 11.816 | 11.547 | 11.434 | 12.370 |
| 117 | 299 | 10.932 | 10.636 | 10.553 | 11.390 | 120 | 292 | 13.031 | 13.292 | 13.082 | 13.310 |
| 117 | 301 | 10.679 | 10.357 | 10.309 | 11.540 | 120 | 293 | 12.786 | 13.165 | 12.883 | 13.240 |
| 118 | 290 | 12.197 | 12.433 | 12.251 | 12.410 | 120 | 294 | 12.641 | 13.038 | 12.714 | 13.070 |
| 118 | 291 | 11.948 | 12.303 | 12.047 | 12.220 | 120 | 295 | 12.577 | 12.910 | 12.574 | 13.100 |
| 118 | 292 | 11.799 | 12.171 | 11.875 | 12.010 | 120 | 296 | 12.563 | 12.782 | 12.455 | 13.190 |
| 118 | 293 | 11.731 | 12.039 | 11.730 | 12.020 | 120 | 297 | 12.562 | 12.652 | 12.353 | 13.020 |
| 118 | 295 | 11.708 | 11.773 | 11.501 | 11.700 | 120 | 298 | 12.550 | 12.522 | 12.258 | 12.900 |
| 118 | 296 | 11.692 | 11.638 | 11.402 | 11.560 | 120 | 299 | 12.512 | 12.391 | 12.166 | 13.185 |
| 119 | 291 | 12.617 | 12.865 | 12.669 | 12.870 | 120 | 300 | 12.446 | 12.259 | 12.072 | 13.287 |
| 119 | 293 | 12.222 | 12.607 | 12.297 | 12.510 |
4. Summary
In this work, we put forward an improved formula to systematically calculate the α-decay energy based on the liquid-drop model. The calculated results are in good agreement with the experimental data. The corresponding average and standard deviation are 0.141 and 0.190 MeV, respectively. The α-decay energy calculated by our improved formula is found to be in better agreement with experimental data compared with the DZ formula and DR formula. Meanwhile, the results calculated by our improved formula are largely consistent with the WS3+ nuclear mass model. Finally, we extend this formula to predict the α-decay energies for nuclei with Z = 117, 118, 119, and 120. The predicted results of these formulae are basically consistent with each other. More importantly, this formula can be used as an independent tool to quantify the α-decay energy of unknown nuclei and further test theoretical predictions of other methods and/or models. It is hoped that this work will be the basis for better experimental design in the future.