α-decay half-life screened by electrons

α -decay half-life screened by electrons 1 corresp first-author Corresponding author, wanniu_nju@163.com Corresponding author, wanniu_nju@163.com wanniu_nju@163.com 1 1 2 Department of Physics and Key Laboratory of Modern Acoustics, Institute of Acoustics, Nanjing University, Nanjing 210093, China Department of Physics and Key Laboratory of Modern Acoustics, Institute of Acoustics, Nanjing University, Nanjing 210093, China Center of Theoretical Nuclear Physics, National Laboratory of Heavy-Ion Accelerator, Lanzhou 730000, China Center of Theoretical Nuclear Physics, National Laboratory of Heavy-Ion Accelerator, Lanzhou 730000, China 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 1 Since the pioneering work of Gamow in 1928 [ 1 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 2 ], the cluster model [ 3 3 - 5 5 3-5 ], the liquid-drop model [ 6 6 ], and the fission-like model [ 7 7 ]. By combining the two-potential approach (TPA) [ 8 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 9 - 13 13 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 14 - 18 18 14-18 ], in a metal environment [ 19 19 - 22 22 19-22 ], on nuclear decays and reactions at astrophysical energies [ 23 23 , 24 24 ], and in dense astrophysical plasmas and super strong magnetic fields [ 25 25 - 30 30 25-30 ]. Previous research only focuses on the screening effects on α decays in one specific environment. In our recent study [ 31 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 1 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 . 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 11 - 13 13 11-13 ] <math display="block" id="M1"><mrow><msub><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi>ℏ</mi><mi>ln</mi><mn>2</mn></mrow><mrow><msub><mi>P</mi><mi>α</mi></msub><mi>F</mi><mfrac><mrow><msup><mi>ℏ</mi><mn>2</mn></msup></mrow><mrow><mn>4</mn><mi>μ</mi></mrow></mfrac><mi>exp</mi><mrow><mo>[</mo><mrow><mo>−</mo><mn>2</mn><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><msub><mi>R</mi><mn>2</mn></msub></mrow><mrow><msub><mi>R</mi><mn>3</mn></msub></mrow></msubsup></mrow></mstyle><mi>d</mi><mi>R</mi><msqrt><mrow><mfrac><mrow><mn>2</mn><mi>μ</mi></mrow><mrow><msup><mi>ℏ</mi><mn>2</mn></msup></mrow></mfrac><mo>|</mo><mi>P</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>|</mo></mrow></msqrt></mrow><mo>]</mo></mrow></mrow></mfrac><mn>,</mn></mrow></math> where P α is the α -particle preformation factor and F is a factor well-defined by the TPA [ 8 8 ]. R 2 and R 3 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 <math display="block" id="M2"><mrow><msub><mi>P</mi><mtext>B</mtext></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>V</mi><mtext>B</mtext></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>Q</mi><mi>B</mi></msub><mn>,</mn></mrow></math> <math display="block" id="M3"><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>V</mi><mrow><mtext>equ</mtext></mrow></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>Q</mi><mi>B</mi></msub><mn>,</mn></mrow></math> where V equ ( R ) is an equivalent interaction potential in an external environment, <math display="block" id="M4"><mtable columnalign="left"><mtr><mtd><msub><mi>V</mi><mrow><mtext>equ</mtext></mrow></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>V</mi><mtext>B</mtext></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>+</mo><mi>Δ</mi><mi>V</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>−</mo><mi>δ</mi><mi>Q</mi></mtd></mtr><mtr><mtd><mo>=</mo><msub><mi>V</mi><mtext>B</mtext></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>+</mo><mi>δ</mi><mi>V</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">[</mo><mi>δ</mi><mi>V</mi><mo>−</mo><mi>δ</mi><mi>Q</mi><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mo>=</mo><msub><mi>V</mi><mtext>B</mtext></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>+</mo><mi>δ</mi><mi>V</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mn>.</mn></mtd></mtr></mtable></math> The quantities Δ V ( R )= δV + δV ( R ) and δQ are the variations of V ( R ) and Q in an external environment, respectively [ 16 16 - 18 18 16-18 ]. The condition δV = δQ presented by Karpeshin and coworker [ 16 16 , 17 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 1 3.1 α decay screened by electrons in neutral atoms 2 In neutral atoms, the variation of the Coulomb potential δV ( R ) is analytically derived in Ref. [ 18 18 ], <math display="block" id="M5"><mrow><mi>δ</mi><mi>V</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>2</mn><mo stretchy="false">(</mo><mn>2</mn><mi>Z</mi><msup><mo stretchy="false">)</mo><mrow><mn>2</mn><mi>γ</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mn>2</mn><mi>γ</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo><mi>γ</mi></mrow></mfrac><mfrac><mrow><msup><mi>e</mi><mn>2</mn></msup></mrow><mrow><msub><mi>a</mi><mn>0</mn></msub></mrow></mfrac><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mi>R</mi><mrow><msub><mi>a</mi><mn>0</mn></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mi>γ</mi></mrow></msup><mn>,</mn></mrow></math> where a 0 is the Bohr radius and the factor <math id="M6"><mrow><mi>γ</mi><mo>=</mo><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>β</mi><mn>2</mn></msup><msup><mi>Z</mi><mn>2</mn></msup></mrow></msqrt></mrow></math> . β = e 2 / 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 19 ]: <math display="block" id="M7"><mrow><mi>δ</mi><mi>Q</mi><mo>=</mo><mi>B</mi><mo stretchy="false">(</mo><mi>Z</mi><mn>,</mn><mi>Z</mi><mo stretchy="false">)</mo><mo>−</mo><mi>B</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>−</mo><mn>2,</mn><mi>Z</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo>−</mo><mi>B</mi><mo stretchy="false">(</mo><mn>2,2</mn><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where B ( Z , Z ) denotes the electron binding energy of an atom with Z protons and Z electrons [ 19 19 ] and the value is given in Ref. [ 32 32 ]. For neutral atoms, the decay energies, Q , are given in the atomic mass evaluation [ 33 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 2 In a metal environment, the variation of δV ( R ) can be divided into two parts [ 18 18 ], <math display="block" id="M8"><mrow><mi>δ</mi><mi>V</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>=</mo><mi>δ</mi><msub><mi>V</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>+</mo><mi>δ</mi><msub><mi>V</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where δV 1 ( R ) is the same as in Eq. (5) from electrons of the mother nucleus and δV 2 ( R ) comes from the metal [ 18 18 ], <math display="block" id="M9"><mrow><mi>δ</mi><msub><mi>V</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>8</mn><msup><mi>e</mi><mn>2</mn></msup></mrow><mi>π</mi></mfrac><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><msub><mi>q</mi><mtext>F</mtext></msub></mrow></msubsup></mrow></mstyle><mtext>d</mtext><mi>q</mi><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><mi>q</mi><mi>R</mi></mrow></msubsup></mrow></mstyle><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>y</mi></msubsup></mrow></mstyle><mtext>d</mtext><mi>x</mi><mrow><mo>[</mo><mrow><msup><mi>F</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><msubsup><mi>q</mi><mtext>F</mtext><mn>2</mn></msubsup></mrow><mrow><mn>3</mn><msup><mi>q</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>]</mo></mrow><mn>,</mn></mrow></math> where F ( x ) is the radial function [ 18 18 ] and the Fermi vector, q F , is determined by the average electron density, n 0 [ 18 18 ], <math display="block" id="M10"><mrow><msub><mi>q</mi><mtext>F</mtext></msub><mo>=</mo><mo stretchy="false">(</mo><mn>3</mn><msup><mi>π</mi><mn>2</mn></msup><msub><mi>n</mi><mn>0</mn></msub><msup><mo stretchy="false">)</mo><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mn>.</mn></mrow></math> Here we take the metal copper (Cu) as an example with n 0 =8.48×10 22 cm -3 . 3.3 α decay screened by electrons in an extremely strong magnetic-field environment 2 In an extremely strong magnetic-field environment, one usually introduces the function ϕ ( x ) to obtain the screened Coulomb potential <math id="M11"><mrow><msub><mi>V</mi><mtext>C</mtext></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mi>Z</mi><mn>1</mn></msub><msub><mi>Z</mi><mn>2</mn></msub><msup><mi>e</mi><mn>2</mn></msup></mrow><mi>R</mi></mfrac><mi>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> [ 26 26 - 29 29 26-29 ]. The function ϕ ( x ) fulfills the equation <math id="M12"><mrow><mfrac><mrow><msup><mtext>d</mtext><mn>2</mn></msup><mi>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mtext>d</mtext><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mi>ϕ</mi><msup><mo stretchy="false">)</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math> [ 25 25 - 30 30 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 1/5 b -2/5 a 0 [ 27 27 - 30 30 27-30 ]. b = B / B 0 is a dimensionless strength [ 26 26 ], where B is the strength in the environment and <math id="M13"><mrow><msub><mi>B</mi><mn>0</mn></msub><mo>=</mo><msubsup><mi>m</mi><mtext>e</mtext><mn>2</mn></msubsup><msup><mi>e</mi><mn>3</mn></msup><mi>c</mi><mo>/</mo><msup><mi>ℏ</mi><mn>3</mn></msup><msup><mrow><mo>=</mo><mn>2.3505</mn><mo>×</mo><mn>10</mn></mrow><mn>9</mn></msup></mrow></math> G is the typical value in neutron stars [ 26 26 ]. The first few terms can be found by applying Baker’s small- x expansion [ 34 34 ] <math display="block" id="M14"><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>+</mo><mi>S</mi><mi>x</mi><mo>+</mo><mfrac><mn>4</mn><mrow><mn>15</mn></mrow></mfrac><msup><mi>x</mi><mrow><mn>2.5</mn></mrow></msup><mo>+</mo><mfrac><mn>2</mn><mrow><mn>35</mn></mrow></mfrac><mi>S</mi><msup><mi>x</mi><mrow><mn>3.5</mn></mrow></msup><mo>−</mo><mfrac><mn>1</mn><mrow><mn>126</mn></mrow></mfrac><msup><mi>S</mi><mn>2</mn></msup><msup><mi>x</mi><mrow><mn>4.5</mn></mrow></msup><mn>,</mn></mrow></math> where S = ϕ ’ (0)=-0.938966 [ 27 27 , 29 29 ]. Then the two parts of Δ V ( R ) can be expressed as follows <math display="block" id="M15"><mrow><mi>δ</mi><mi>V</mi><mo>=</mo><mfrac><mrow><msub><mi>Z</mi><mn>1</mn></msub><msub><mi>Z</mi><mn>2</mn></msub><msup><mi>e</mi><mn>2</mn></msup></mrow><mi>R</mi></mfrac><mi>S</mi><mi>x</mi><mn>,</mn></mrow></math> <math display="block" id="M16"><mrow><mi>δ</mi><mi>V</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mi>Z</mi><mn>1</mn></msub><msub><mi>Z</mi><mn>2</mn></msub><msup><mi>e</mi><mn>2</mn></msup></mrow><mi>R</mi></mfrac><mo stretchy="false">[</mo><mfrac><mn>4</mn><mrow><mn>15</mn></mrow></mfrac><msup><mi>x</mi><mrow><mn>2.5</mn></mrow></msup><mo>+</mo><mfrac><mn>2</mn><mrow><mn>35</mn></mrow></mfrac><mi>S</mi><msup><mi>x</mi><mrow><mn>3.5</mn></mrow></msup><mo>−</mo><mfrac><mn>1</mn><mrow><mn>126</mn></mrow></mfrac><msup><mi>S</mi><mn>2</mn></msup><msup><mi>x</mi><mrow><mn>4.5</mn></mrow></msup><mo stretchy="false">]</mo><mn>.</mn></mrow></math> 4 Numerical results and discussion 1 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 , <math display="block" id="M17"><mrow><msub><mi>Δ</mi><mrow><mtext>sc</mtext></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>T</mi><mrow><mtext>sc</mtext></mrow></msub><mo>−</mo><msub><mi>T</mi><mrow><mtext>nsc</mtext></mrow></msub></mrow><mrow><msub><mi>T</mi><mrow><mtext>nsc</mtext></mrow></msub></mrow></mfrac><mn>,</mn></mrow></math> which includes Δ Atom , Δ Metal , and Δ Mag , corresponding to neutral atoms, a metal, and a magnetic-field environment. In Fig. 2 Fig. 2 , the variation Δ sc is given with Fig. 2(a) Fig. 2(a) for neutral atoms, Fig. 2(b) Fig. 2(b) for a metal environment, and Fig. 2(c)-2(f) Fig. 2(c)-2(f) all for a magnetic environment, but with different strengths: b =10 3 (c), b =10 4 (d), b =10 5 (e), and b =10 6 (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 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) 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 . Besides, in each chart of Fig. 2 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 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 159 Lu 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. 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 235 U in Fig. 4 Fig. 4 . As shown in Fig. 4 Fig. 4 , if only Δ V ( 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. 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(Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2016 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-016-0150-2 http://dx.doi.org/10.1007/s41365-016-0150-2 . 2016 Nuclear Science and Techniques 06 27