Niu Wan·Chang Xu·Zhong-Zhou Ren,2

α-Decay half-life screened by electrons

Niu Wan1·Chang Xu1·Zhong-Zhou Ren1,2

In this paper,by considering the electrons in different external environments,including neutral atoms,a metal,and an extremely strong magnetic-field environment, the screened α-decay half-lives of the α emitters with proton number Z=52-105 are systematically calculated.In the external environment,the decay energy and the interaction potential between α particle and daughter nucleus are both changed due to the electron screening effect and their variations are both very important for the electron screening effect.Besides,the electron screening effect is found to be closely related to the decay energy and its proton number.

Electron screening effect.α-Decay half-life. Density-dependent cluster model

1 Introduction

Since the pioneering workof Gamow in 1928[1]where α decay was successfully explained as a quantum tunneling effect,much attention has been paid to the α decays of unstable nuclei and several analyses have been performed to calculate the half-lives of α emitters throughout the nuclide chart with the shell model[2],the cluster model[3-5],the liquid-drop model[6],and the fission-like model[7].By combining the two-potential approach(TPA)[8]and a microscopicpotential,weinvestigatedtheα-decayhalf-lives of both spherical and deformed nuclei by using the densitydependent cluster model(DDCM)[9-13].

Although lots of theoretical studies of α decays have been conducted,α-decayhalf-lifescreenedbyelectronshasnotbeen systematically studied.The electron screening effect is discussed only in a few theoretical works in several external environments,such as in neutral atoms within different approaches[14-18],in a metal environment[19-22],on nucleardecaysandreactionsatastrophysicalenergies[23,24], andin dense astrophysical plasmas and super strongmagnetic fields[25-30].Previousresearchonlyfocusesonthescreening effects on α decays in one specific environment.In our recent study[31],the screened α-decay half-lives are systematically calculated with the DDCM in external environments,namely neutralatoms,ametal,andanextremelystrongmagnetic-field environment.A brief review is given here.

2 General analysis of α decays screened by electrons

In an external environment,the interaction potential, V(R),between α particle and daughter nucleus and the decay energy,Q are both changed,resulting in a variation in the potential barrier that the α particle penetrates,as shown in Fig.1.

It is obvious that the screened potential barrier is different from the non-screened(bare)one so that the α-decay half-life T1/2will be changed in an external environment. In the DDCM,T1/2can be calculated by[11-13]

where Pαis the α-particle preformation factor and F is a factor well defined by the TPA[8].R2and R3are two turning points.P(R)is defined as P（R）=V（R）-Q,where V(R)is the interaction potential and Q is the decay energy. For a bare nucleus and in an external environment,P(R) can be expressed as

where Vequ（R）is an equivalent interaction potential in an external environment,

The quantities ΔV（R）=δV+δV（R）and δQ are the variations in V(R)and Q in an external environment, respectively[16-18].The condition δV=δQ presented by Karpeshin et al.[16,17]is also applied here.Equations(2-4)show that the term δV（R）results in the difference of the α-decay half-life between the bare nucleus and the external environment.In the following section,we will present the terms δV（R）in different external environments.

Fig.1 Comparison of non-screened and screened potential barriers that α particle penetrates

3 α decay in external environments

3.1 α decay screened by electrons in neutral atoms

In neutral atoms,the variation in the Coulomb potential δV（R）is analytically derived in Ref.[18],

The variation δQ can be obtained from the difference of the electron binding energies of the three particles[19]:

where B(Z,Z)denotes the electron binding energy of an atom with Z protons and Z electrons[19]and the value is given in Ref.[32].For neutral atoms,the decay energies, Q,are given in the atomic mass evaluation[33].So the decay energies for bare nuclei can be calculated by QB=Q-δQ.Then the half-lives for bare nuclei and neutral atoms can be calculated.

3.2 α decay screened by electrons in a metal environment

In a metal environment,the variation in δV（R）can be divided into two parts[18],

where δV1（R）is the same as in Eq.(5)from electrons of the mother nucleus and δV2（R）comes from the metal[18],

where F(x)is the radial function[18]and the Fermi vector, qF,is determined by the average electron density,n0[18],

Here we take the metal copper(Cu)as an example with n0=8.48×1022cm-3.

3.3 α decay screened by electrons in an extremely strong magnetic-field environment

where S=φ'（0）=-0.938966[27,29].Then the two parts of ΔV（R）can be expressed as follows

4 Numerical results and discussion

By applying the DDCM,we perform systematic calculations of the electron-screened α-decay half-lives of nuclei with the proton number Z=52-105 in different external environments.Here we only consider the favored α transitions to avoid the uncertainties coming from the nonzero angular momentum.The difference between the screened α-decay half-life,Tsc,and non-screened one,Tnsc,is defined by Δsc,

which includes ΔAtom，ΔMetal,and ΔMag,corresponding to neutral atoms,a metal,and a magnetic-field environment.

Fig.2 Relative variation Δsc=（Tsc-Tnsc）/Tnscin neutral atoms(a), in metal(b),and in a magnetic-field environment with b=103(c), b=104(d),b=105(e),and b=106(f)

In Fig.2,the variation Δscis given in Fig.2a for neutral atoms,Fig.2b for a metal environment,and Fig.2c-f all for a magnetic environment,but with different strengths: b=103(c),b=104(d),b=105(e),and b=106(f).It can be seen that Δscvalues are all positive.So the α-decay half-lives are all increased by the electrons in external environments because of slightly higher potential barrier as shown in Fig.1.Then the α-particle penetration probability is relatively smaller compared to bare nuclei,leading to longer α-decay half-life.For neutral atoms and a metal environment in Fig.2a,b,the screened α-decay half-life is varied moderately,but in a magnetic-field environment the variation can be very large,and depends closely on the strength b.

Besides,in each chart of Fig.2 there are several significantly larger Δscalong an isotopic chain.We find that these values are closely related to the small decay energies, Q.In Fig.3,we plot their correlation for a typical isotopic chain,Lu.It is clearly seen that the variation in Δscdecreases with Q and the smallest decay energy of159Lu is corresponding to the biggest Δsc.This is because the electron screening effects are approximately the same for all Lu isotopes.Thus Δscmainly depends on Q.The variation Δscis also related to the proton number.To measure electron screening effects in experiments,α-decay candidates with relatively small decay energies and proper decay half-lives are suggested.

In a magnetic-field environment,the variation in Δsccan be very large and increases with the strength,b.Thus this environment could have a significant effect on α decays.To show the details,we plot the decay half-life ratio fM= TMag/Tnscfor235U in Fig.4.As shown in Fig.4,if only ΔV（R）is considered,the ratio,fM,decreases sharply with b.Oppositely,the ratio,fM,increases sharply with b if only δQ is considered.However,when both are included,the ratio,fM,still increases with b,but the increase is much slower.Thus the variations in V(R)and Q compete with each other and both are important factors for the electron screening effect.

Fig.3 Relation between Δscand the decay energy,Q,for a typical isotopic chain of Z=71

Fig.4 Screened and non-screened half-lives ratio fM=TMag/Tnscfor 235U with different magnetic-field strengths,b

5 Summary

With the DDCM,the electron-screened α-decay half-life has been systematically calculated in external environments,including neutral atoms,a metal,and an extremely strong magnetic-field environment.From the numerical results,it can be concluded that the electron screening effects on α decays in neutral atoms and in a metal environment are very moderate.But in magnetic-field environments,the effect depends closely on the field strength. Besides,both the variations in the interaction potential and the decay energy are important for the electron screening effect.Similarly to previous studies,the electron screening effect is also closely related to the decay energy and the proton number.

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11 August 2016/Revised:2 October 2016/Accepted:9 October 2016

©Shanghai Institute of Applied Physics,Chinese Academy of Sciences,Chinese Nuclear Society,Science Press China and Springer Science+Business Media Singapore 2016

This work was supported by the National Natural Science Foundation of China(Nos.11175085,11235001,11375086,and 11120101005), by the 973 Program of China(No.2013CB834400),by the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions(PAPD),and by the Open Project Program of the State Key Laboratory of Theoretical Physics,Institute of Theoretical Physics,Chinese Academy of Sciences,China(No. Y5KF141CJ1).

✉ Niu Wan

wanniu_nju@163.com

1Department of Physics,Key Laboratory of Modern Acoustics,Institute of Acoustics,Nanjing University, Nanjing 210093,China

2Center of Theoretical Nuclear Physics,National Laboratory of Heavy-Ion Accelerator,Lanzhou 730000,China