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Let’s consider the EC process,
The EC decay probability per time unit is given by Fermiis golden rule,
where
$ i $ ,$ f $ are initial and final states respectively,$ \rho_f $ is the neutrino final states per energy unit, and$ \hat{O} $ is the weak interaction operator.In the initial state, there are one proton, one electron, and other nucleons in
$ ^A_Z X $ ,$ \left| i \right\rangle = \left| {p,e} \right\rangle \left| {_{Z - 1}^{A - 1}Y} \right\rangle $ . In the finial state, there have one neutron, one neutrino, and other nucleons in$ ^{\,\,\,\,\,\,A}_{Z-1}Y $ ,$ \left| f \right\rangle = \left| {n,\nu } \right\rangle \left| {_{Z - 1}^{A - 1}Y} \right\rangle $ . Since the operator$ \hat{O} $ acts only on weak interaction participants, the EC decay rate is roughly proportional to the possibility of finding the electron in the nuclear volume[19],where
$ \psi_{\rm e} $ is the electron's wave function, and$ r_{\rm n} $ is radius of the nucleus. Here we use the assumption that the nucleon's wave function is constant in the nuclear volume. Furthermore, the coulomb interaction difference between the normal Bohr state and the DDL is very small compared with the strong interaction inside the nuclei. Therefore, the other parts of the matrix for the Bohr level and the DDL are roughly same. Defining$ T_{1/2} $ to be the half life of capturing an electron from normal Bohr level, and$ T_{1/2}^{\#} $ from DDL, their ratio can be simplified as[19-20]:where the
$ Q^{} $ and$ Q^{\#} $ is the reaction Q-value for Bohr and the DDL states respectively. The$ \frac{|\psi_0^{}(r_{\rm n})|}{|\psi_0^{\#}(r_{\rm n})|} $ is the ratio of the electron wave functions evaluated at the nuclear surface$ r = r_{\rm n} $ . And also take$ Q^{\#} = Q-m_0c^2 $ .From Eq. (7) and (10), we have
Since
$ r_{\rm n} $ is in order of fm level,$ r_0^\# $ is about 390 fm, and$ r_0 $ is about 0.53$ \mathop {\rm{A}}\limits^ \circ $ , we have$ \exp[{r_{\rm n}/r_0^{\#}-r_{\rm n}/r_0}]\simeq 1 $ . The equation can be re-written as,insert it to the Eq. (17), we have
The results, chosen as examples, are shown as
$ T_{1/2}^{\rm{(DDL1)}} $ in Table 1.Nucleus $ Q $/MeV Z $ T_{1/2}^{(0)} $ $ T_{1/2}^{\rm{(DDL1)}} $ $ T_{1/2}^{\rm{(DDL2)}} $ 7Be 0.861 4 53.2 d 21 ms 52 ms 11C 1.982 6 20.3 m 18 μs 19 μs 13N 2.220 7 10.0 m 14 μs 16 μs 15O 2.757 8 122 s 4.3 μs 4.8 μs 23Mg 4.056 12 11.3 s 1.5 μs 1.9 μs 30P 4.232 15 2.50 m 47 μs 64 μs 53Fe 3.742 26 8.51 m 1.3 ms 2.9 ms 62Cu 3.958 29 9.67 m 2.2 ms 5.9 ms 63Zn 3.367 30 38.47 m 10 ms 30 ms 64Cu 1.675 29 12.7 h 0.28 s 0.7 s Table 1. The new life time of several radioactive nuclei if the DDL exist.
$ T_{1/2}^{(0)} $ represents the original half life of the corresponding nucleus.We also numerically solve the Eq. (1) with a more realistic potential, specifically, assuming that the charge is evenly distributed in the nucleus. The potential U in Eq. (1) has the form:
Once obtaining the numerical wave functions, the EC decay ratio were then calculated according to Eq. (15). The results are shown as
$ T_{1/2}^{\rm{(DDL2)}} $ in Table 1. As an example, the numerically-solved wave functions for 62Cu's Bohr state and DDL are shown in Fig. 1.Figure 1. A comparison of the numerical evaluation of
$4\pi r^2|\psi(r)|^2 (r < r_{\rm n})$ for the 62Cu's Bohr state (dash line) and DDL (solid line).The present studies reveal that the nuclei listed in the Table 1 are possible candidates for the laser DDL studies. They will be prepared at accelerators or nuclear reactors in advance, and then server as laser targets. They can be obtained by one-nucleon transferring reactions and relatively easier to be prepared. Furthermore, because theirs
$ T_{1/2}^{\rm{(DDL)}} $ are much longer than 100 ns, but relatively shorter regarding the cosmic and ambient radiation backgrounds (look the argument in Sec. 3), a high signal-to-noise ratio can be achieved if using these nuclei.
Feasibility Study on the Deep Dirac Levels with High-Intensity Lasers
doi: 10.11804/NuclPhysRev.37.2019CNPC20
- Received Date: 2019-12-25
- Rev Recd Date: 2020-04-10
- Available Online: 2020-09-30
- Publish Date: 2020-09-20
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Key words:
- Deep Dirac Level /
- electron capture life time /
- high intensity laser /
- manipulating nuclear decay life time
Abstract: Various theories have predicted the deep Dirac levels (DDLs) in atoms for many years. However, the existence of the DDL is still under debating, and need to be confirmed. With the development of high intensity lasers, nowadays, electrons can be accelerated to relativistic energies by high intensity lasers. Furthermore, electron-positron pairs can be created, and nuclear reactions can be ignited, which provide a new tool to explore the DDL related fields. In this paper, we propose a new experimental method to study the DDL levels by monitoring nuclei's orbital electron capture life time in plasma induced by high intensity lasers. The present study reveal that if a DDL exists, a nuclear electron capture rate could be enhanced by factor of over
Citation: | Changbo FU, Xiaopeng ZHANG, Dechang DAI. Feasibility Study on the Deep Dirac Levels with High-Intensity Lasers[J]. Nuclear Physics Review, 2020, 37(3): 377-381. doi: 10.11804/NuclPhysRev.37.2019CNPC20 |