α-decay half-life screened by electrons

Special Section on International Workshop on Nuclear Dynamics in Heavy-Ion Reactions (IWND2016)

α-decay half-life screened by electrons

Niu Wan，

Chang Xu，

Zhong-Zhou Ren

Nuclear Science and Techniques

Vol.27, No.6

Article number 149

Published in print 20 Dec 2016

Available online 31 Oct 2016

DOI：10.1007/s41365-016-0150-2

48400

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 effectα-decay half-lifeDensity-dependent cluster model

1 Introduction

Since the pioneering work of 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 microscopic potential, we investigated the α-decay half-lives of both spherical and deformed nuclei by using the density-dependent cluster model (DDCM) [9-13].

Although lots of theoretical studies of α decays have been conducted, α-decay half-life screened by electrons has not been 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 nuclear decays and reactions at astrophysical energies [23, 24], and in dense astrophysical plasmas and super strong magnetic fields [25-30]. Previous research only focuses on the screening 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, neutral atoms, a metal, and an extremely strong magnetic-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 of the potential barrier that the α particle penetrates, as shown in Fig. 1.

Fig. 1.

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

It is obvious that the screened potential barrier is different from the non-screened (bare) one so that the α-decay half-life T_1/2 will be changed in an external environment. In the DDCM, T_1/2 can be calculated by [11-13]

T_{1 / 2} = \frac{ℏ \ln 2}{P_{α} F \frac{ℏ^{2}}{4 μ} \exp [- 2 \int_{R_{2}}^{R_{3}} d R \sqrt{\frac{2 μ}{ℏ^{2}} | P (R) |}]},

(1)

where Pα is the α-particle preformation factor and F is a factor well-defined by the TPA [8]. R₂ and R₃ are 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

P_{B} (R) = V_{B} (R) - Q_{B},

(2)

P (R) = V_{equ} (R) - Q_{B},

(3)

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

\begin{array}{l} V_{equ} (R) = V_{B} (R) + Δ V (R) - δ Q \\ = V_{B} (R) + δ V (R) + [δ V - δ Q] \\ = V_{B} (R) + δ V (R) . \end{array}

(4)

The quantities ΔV(R)=δV+δV(R) and δQ are the variations of V(R) and Q in an external environment, respectively [16-18]. The condition δV=δQ presented by Karpeshin and coworker [16, 17] is also applied here. It can be seen from Eqs. (2)-(4) 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.

3 α decay in external environments

3.1 α decay screened by electrons in neutral atoms

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

δ V (R) = \frac{2 (2 Z)^{2 γ + 1}}{Γ (2 γ + 2) γ} \frac{e^{2}}{a_{0}} {(\frac{R}{a_{0}})}^{2 γ},

(5)

where a₀ is the Bohr radius and the factor $γ = \sqrt{1 - β^{2} Z^{2}}$ . β=e²/ħc is the fine-structure constant.

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

δ Q = B (Z, Z) - B (Z - 2, Z - 2) - B (2,2),

(6)

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 Q_B=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 of δV(R) can be divided into two parts [18],

δ V (R) = δ V_{1} (R) + δ V_{2} (R),

(7)

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

δ V_{2} (R) = \frac{8 e^{2}}{π} \int_{0}^{q_{F}} d q \int_{0}^{q R} \frac{d y}{y^{2}} \int_{0}^{y} d x [F^{2} (x) - \frac{x^{2} q_{F}^{2}}{3 q^{2}}],

(8)

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

q_{F} = (3 π^{2} n_{0})^{1 / 3} .

(9)

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

In an extremely strong magnetic-field environment, one usually introduces the function ϕ(x) to obtain the screened Coulomb potential $V_{C} (R) = \frac{Z_{1} Z_{2} e^{2}}{R} ϕ (x)$ [26-29]. The function ϕ(x) fulfills the equation $\frac{d^{2} ϕ (x)}{d x^{2}} = (x ϕ)^{1 / 2}$ [25-30] with two boundary conditions: ϕ(0)=1 and ϕ’(0)=-0.938966, where x=R/R_s is the screening factor with R_s=1.041863Z^1/5b^-2/5a₀ [27-30]. b=B/B₀ is a dimensionless strength [26], where B is the strength in the environment and $B_{0} = m_{e}^{2} e^{3} c / ℏ^{3} {= 2.3505 \times 10}^{9}$ G is the typical value in neutron stars [26]. The first few terms can be found by applying Baker’s small-x expansion [34]

ϕ (x) = 1 + S x + \frac{4}{15} x^{2.5} + \frac{2}{35} S x^{3.5} - \frac{1}{126} S^{2} x^{4.5},

(10)

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

δ V = \frac{Z_{1} Z_{2} e^{2}}{R} S x,

(11)

δ V (R) = \frac{Z_{1} Z_{2} e^{2}}{R} [\frac{4}{15} x^{2.5} + \frac{2}{35} S x^{3.5} - \frac{1}{126} S^{2} x^{4.5}] .

(12)

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 non-zero angular momentum. The difference between the screened α-decay half-life, T_sc, and non-screened one, T_nsc, is defined by Δ_sc,

Δ_{sc} = \frac{T_{sc} - T_{nsc}}{T_{nsc}},

(13)

which includes Δ_Atom, Δ_Metal, and Δ_Mag, corresponding to neutral atoms, a metal, and a magnetic-field environment.

In Fig. 2, the variation Δ_sc is given with Fig. 2(a) for neutral atoms, Fig. 2(b) for a metal environment, and Fig. 2(c)-2(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 Δ_sc values 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 Figs. 2(a) and 2(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.

Fig. 2.

The relative variation Δ_sc=(T_sc-T_nsc)/T_nsc in 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).

Besides, in each chart of Fig. 2 there are several significantly larger Δ_sc along 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 Δ_sc decreases with Q and the smallest decay energy of ¹⁵⁹Lu is corresponding to the biggest Δ_sc. This is because the electron screening effects are approximately the same for all Lu isotopes. Thus Δ_sc mainly depends on Q. The variation Δ_sc is 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.

Fig. 3.

The relation between Δ_sc and the decay energy, Q, for a typical isotopic chain of Z=71.

In a magnetic-field environment, the variation in Δ_sc can 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 f_M=T_Mag/T_nsc for ²³⁵U in Fig. 4. As shown in Fig. 4, if only ΔV(R) is considered, the ratio, f_M, decreases sharply with b. Oppositely, the ratio, f_M, increases sharply with b if only δQ is considered. However, when both are included, the ratio, f_M, still increases with b, but the increase is much slower. Thus the variations of V(R) and Q compete with each other and both are important factors for the electron screening effect.

Fig. 4.

The screened and non-screened half-lives ratio f_M=T_Mag/T_nsc for 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 of 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.

References

[1]

G. Gamow,

Zur Quantentheorie des Atomkernes

. Z. Phys. 51, 204-212 (1928). doi: 10.1007/BF01343196