Nuclear isomers, being meta-stable states of nuclei, have a profound impact on various fields including nuclear structure models [1,2], nuclear astrophysics [2,3], nuclear lasers [4], nuclear batteries [5], nuclear clocks [6], fine structure measurement [7], and more [8–10]. In order to advance applications in the field of nuclear isomers, it is crucial to understand the mechanisms for exciting or de-exciting them. How to effectively excite the nucleus from the ground state to the isomeric state involves Coulomb excitation (CE) [11], photoabsorption [12], multi-photon excitation [13], and atomic processes related to electron-nucleus interaction. The latter includes nuclear excitation by electron capture (NEEC) [14], nuclear excitation by electronic transition (NEET) [15] and electron bridge (EB) [16]. Some of the more in-depth studies, such as CE and direct γ-photon absorption, their excitation probability is relatively low [17]. Therefore, the potential effective nuclear excitation mechanism of electron coupling in atoms has been considered, especially the NEEC process. NEEC is the inverse process of the internal conversion (IC). The concept of NEEC was first proposed by V.I. Goldanskii et al. in 1976 [14,17]. The principle of NEEC is shown in Figure 1A. Similar to the excitation of atoms in dielectronic recombination [18], where an ion captures a free electron leading to the excitation of the atom, the NEEC is a process in which an ion captures a free electron leading to the resonance excitation of atomic nuclei, which can then be de-excited by emitting γ-ray or IC electrons. For most nuclei, their NEEC resonance cross sections are much larger than the direct photon absorption cross sections, which makes the NEEC to be a promising mechanism for effective nuclear excitation [17]. Just like the “electron captured on nuclei” process [19,20], the NEEC process, which is an “electron captured on atoms” process, play a significant role in the nucleosynthesis in astrophysical environments [21–24], such as in stars like the sun, nova, or supernova explosions.

Considerable effort has been devoted to describing the NEEC process theoretically, as well as verifying it experimentally, which include methods of using laser-generated plasma [14,25–30], storage ring [17,31,32], channeling through crystals [33], beam-based scenario [34–36], and so on. In 2018, C. J. Chiara et al. claimed to have discovered the first experimental evidence of NEEC in the isomer depletion of ^93mMo in a beam-based scenario [34]. However, this result was quickly called into questions by subsequent literature for overestimating the theoretical NEEC cross section [35,37] and underestimating the experimental backgrounds [38]. Recently, the isomer beam experiment overturned the previous experimental results [39], but the estimates of the upper limits of their yields remain consistent with the results of theoretical calculations [35,37]. Consequently, the urgent need for experimental confirmation of the NEEC process persists to this day.

In this work, we propose a new experimental scheme for verifying the existence of the NEEC process based on electron beam ion traps (EBIT). The EBIT is a type of ion trap that uses electron beams to trap and confine ions. Traditionally, the EBIT has been used in various fields, from basic atomic and molecular physics to plasma physics [40,41]. By adding an extra electron gun, the modified EBIT is expected to be used to study the nuclear physical processes like the NEEC.

The remaining sections are organized as follows: Section 2 focuses on the theoretical method of calculating the NEEC cross sections for charged ions [31,36,42]. To detect the NEEC process, Section 3 proposes an experimental setup based on an EBIT. The numerical simulation results are provided in Section 4, demonstrating the possibility of NEEC detection. Section 5 presents a potential error budget based on the proposed experimental setup. Lastly, Section 6 provides a summary of this study.

The NEEC is a nuclear excitation process in which a free electron is captured by the atom. Its Feynman diagram is shown in Figure 1B, where the original electron e is at continuum state, and the final electron is marked as “ⓔ” to indicate that it is atomically bounded. By transiting the energy released from the free electron, a nuclear isomer could be produced through this process. Because of the energy conservation, one has E _r = E _n − E _b , where E _r is the kinetic energy carried by the free electron, E _b is the binding energy to which the free electron is captured, and E _n is the nuclear excitation energy.

It has been shown theoretically that the cross sections of the NEEC process normally are much higher than the photon absorption cross sections [31,36,42], which makes NEEC a good method to produce nuclear isomers. Furthermore, the nuclear isomer cross sections could be further enhanced if the free electron is captured to a highly charged state. In fact, as shown in Ref. [42], for a certain final electronic configuration α r ( n l j ) and ion charge state q before the electron capture, the NEEC cross section for a nucleus in its ground state initially is written as, σ NEEC q , α r E e = K λ e 2 2 α IC q , α r Γ γ L r E e − E r , (1) where K = 2 J E + 1 2 j c + 1 2 J G + 1 2 j f + 1 , (2) J _E and J _G are the nuclear spins of the excited and ground states, respectively; j _c and j _f are the total angular momenta of the captured and free electrons, respectively [22]. λ _e is the free electron wavelength, and Γ _γ is the nuclear transition width of the electromagnetic radiation. The partial IC coefficients (ICCs) α IC q , α r , depends on the α _r and q. L r E e − E r is a Lorentzian function centered at the resonance energy E _r . The integral of the NEEC cross section σ NEEC q , α r ( E e ) over the free projectile electron energy E _e is called resonance strength S NEEC q , α r , which can be expressed as S NEEC q , α r = ∫ σ NEEC q , α r E e d E e = K λ e 2 2 α IC q , α r Γ γ . (3)

Assuming a linear scaling dependence, one can deduce the ICCs for ionized atoms ( α IC q , α r ) in Eq. 3 from the ICCs of neutral atoms α I C q = 0 , n l j [36], α IC q , α r = α IC q = 0 , n l j E b q , α r E b q = 0 , n l j n h n max , (4) where E b q , α r and E b q = 0 , n l j are the binding energies for ions in the charge state q and neutral atoms, respectively. Their ratio accounts for the increase of the ICCs with the ionization level [24,43]. The ratio between the present n _h and the maximum n _max number of holes in the capture subshell nl _j accounts for the decrease of the ICCs for partially filled subshells [43,44].

The following steps are taken to calculate α IC q , α r in Eq. 4. To obtain the ICCs of neutral atoms, α I C q = 0 , n l j , the code BrIccFO package is employed [45], which employs the frozen orbital approximation to take into account the effect of the hole. The binding energies E b q = 0 , n l j for neutral atoms are taken from tables in Ref. [46], while the ones for highly ionized atoms, E b q , α r , are taken from the NIST database [47].

Isomeric states of nuclei with relatively low nuclear energy levels are chosen as potential candidates for searching NEEC experimentally, and their NEEC excitation resonance strengths are calculated, as listed in Table 1. It can be seen that for most of the longer-lived ( > 1 ms) isomeric states, the NEEC excitation resonance strengths are relatively low, about 10^–6 order of magnitude below, which may be due to the fact that the transition has a high multi-polarity. The ¹⁸¹Ta⁵⁹⁺, which has the relatively high electric transition strength among all the nuclei considered here, has a strength of 4.34 × 10⁻³ b⋅eV. It is mainly due to its very low γ transition multipolarity (E1), as well as its large ICC ( α IC q = 0,3 p 3 / 2 = 6.72 ) . In fact, we have conducted a detailed study of the resonance strengths of NEEC in ¹⁸¹Ta across various charged states and subshells under the ground state assumption (NEEC-GSA) [42]. Figure 2 shows resonance strengths of NEEC for ¹⁸¹Ta’s M, N, O, and P subshells, calculated as a function of the projectile electron energy E _e and the charge states q (1 ≤ q ≤ 59) of ^181mTa ions. It can be founded that the maximum value of the resonance strength is 6 orders of magnitude larger than the minimum one. The maximum resonance strength is typically observed in the inner shell, which corresponds to a higher charged state. This is due to the fact that the electrons in the inner shell are closer to the nucleus, which increases the likelihood of exciting them, as long as the energy conservation principle is satisfied.

Our calculations show that highly charged ions (HCIs), including ¹⁵⁷Gd, ¹⁶¹Dy, ¹⁸¹Ta, ⁸³Kr, ⁹⁹Ru, ¹²⁹I, ¹³⁸La, ¹⁴⁵Nd, ⁵⁷Fe, ¹¹⁹Sn, ¹³⁷La, ²⁰¹Hg, etc., may be very promising in searching for the NEEC process. If they are ionized to HCIs, they are relatively easy to be excited to isomeric states due to their high NEEC resonance strengths, which are further discussed in detail in the subsequent sections.

As shown in the preceding section, the calculation results indicate that the NEEC resonance strengths can be enhanced significantly if HCIs are used, suggesting their potential in NEEC search experiments. An EBIT is a device that utilizes a monoenergetic, high-current-density electron beam to generate and then confine HCIs. With appropriate modifications, the EBITs have the potential to be valuable instruments for investigating NEEC.

The principle we propose for investigating NEEC utilizing an EBIT is illustrated in Figure 3. In contrast to conventional EBIT configurations that employ only one electron gun, our design incorporates two electron guns with varying energies. The electron beams are extracted from the electron guns and accelerated to designed energies, from tens to hundreds keV. Then the electron beams fly into the drift tubes, bombard, and then ionize the atoms there. The product ions undergo triple confinements. Firstly, they are radially confined by the spatial charge of a dense electron beam. The superconducting magnetic field provides further radial and longitudinal confinements. Further longitudinal confinement is achieved by the static electric field of three electrodes in the drift tubes. As the atoms/ions are continuously bombarded by electron beams, they can be stripped to HCIs. If the energy of the electrons matches the NEEC resonance conditions for the electron-HCI, the nuclei can be excited to their isomeric state. By placing a photon detector, such as a High-purity germanium detector, in close proximity, one can monitor the gamma photons emitted by the excited nuclei.

An electron gun can provide a beam density over 10¹³ electrons ⋅s⁻¹cm⁻², with energy fluctuation of tens of eV today. The HCIs can be confined in the trap with a diameter of tens of micrometers, a length of a few centimeters, and a density in the order of 10⁸ cm⁻³. With the help of an EBIT device, HCIs of almost any element in the periodic table can now be produced.

However, the energy required for stripping does not necessarily match that for the resonance. Therefore, to address this issue, in our proposed setup, we use two electron guns. One gun is tuned to strip the atoms to HCIs, while the other is tuned to match the resonance conditions of the NEEC. As shown in Figure 3, the two electron beams from the two electron guns can be deflected by a dipole magnet, converge to be co-axial, and then pass through the confined HCIs area. As an example, for isotope ¹⁸¹Ta, by tuning the first electron gun to 17.721 keV, one can stripe it to ¹⁸¹Ta⁵⁹⁺ [48]. If tuning the second electron gun to 0.33 keV, the 0.33 keV electrons from the 2nd gun can excite ¹⁸¹Ta⁵⁹⁺ to their isomeric state of (9/2⁻, 6.237 keV) with the cross section of 4.34 × 10⁻³ b⋅eV.

Consider HCIs that are trapped in an EBIT, as shown in Figure 3. It is assumed that the ions are confined within a cylindrical volume with a radius of R and a length of L cm. Additionally, it is assumed that the electron energy spectrum follows a Gaussian distribution with a width of ϵ _e . The spatial spreading of the electron beam is also assumed to be Gaussian with a width of σ _b . The rate of nuclear isomers generated by the NEEC process for a given electron beam in EBIT can be written using the resonance strength Eq. 3 as R NEEC E e = ∫ j e f E e − E r n i σ NEEC E e L d E = j e f E e − E r n i S NEEC L , (5) where j represents the electron beam current intensity, e is the charge of an electron, n _i represents the density of ion trapped in an EBIT, L is the length of the ion cluster in the trap, E _e is the energy of the electron beam, and f (E _e − E _r ) represents the energy distribution of the electron beam.

To ensure practicality, we initially screened candidate nuclei for NEEC studies based on certain criteria. Ideally, we looked for stable isotopes or isotopes with a long lifetime (more than 1 year) to avoid the need for a radioactive ion source. Additionally, we preferred isotopes with relatively lower transition energy (less than 300 keV) since it is hard for typical EBITs to provide electron energies larger than that. We also considered the lifetime of the isomeric state, which should be relatively long but not too long (in the range of tens of nanoseconds to microseconds). A longer lifetime results in a narrower resonance width and smaller reaction yield. However, if the lifetime is too short, it may overlap with the radioactive recombination (RR) process [49,50], which is an atomic process with a short lifetime, typically in order of 10^–14 s [49]. The RR can be a significant background for NEEC studies.

Taking ¹⁸¹Ta as an example, it can be stripped to the 59⁺ valence state, which corresponds to a binding energy of E _b = 5.907 keV. To achieve the highest ratio of expected charge state to all ions, the stripping electron beam energy E _es should be tuned to be 2 ∼ 3 times the ion binding energy E _b , i.e., E _es ≈ (2 ∼ 3)E _b [48]. Ta ions can capture an electron to the 3p _3/2 atomic level (58⁺ charge state) through the NEEC process while being excited from the ground state (J ^π = 7/2⁺, stable) to a 6.237 keV isomeric state ( J π = 9 / 2 − , T 1 / 2 ex = 6.05 μ s ) . Therefore, for the beam serving as the NEEC resonance beam to match the nuclear excitation resonance condition, its beam energy E _er can be tuned to be 0.33 keV.

Consider that ¹⁸¹Ta ions are trapped in a typical EBIT, and the trapped ions fill a cylindrical volume with a radius of 50 μm and a length of 3 cm. We assume that n _i is 10⁸ cm⁻³ in the ion trap and that a resonant electron beam with a current of 200 mA is incident. The electron beam with an energy spread of under 50 eV (ϵ _e ) and a radius of about 50 μm (σ _b ) is achievable today. Normally the nuclear energy level width Γ n th = Γ γ + Γ IC is very narrow compared with the beam energy spread, where Γ_IC represents the IC decay width. For instance, the ¹⁸¹Ta isotope’s total nuclear transition width is approximately Γ n th = 3.27 × 1 0 − 11 eV, compared with the electron beam energy spread of 50 eV. Therefore, we also need to consider the influence of resonance strength. The value of the resonance strength is calculated to be S _NEEC = 4.34 × 10⁻³ b⋅eV at the resonance energy E _r = 0.33 keV.

The calculated rates based on the above assumptions are listed in Table 2. As an order of magnitude estimation, the distribution of charge states is not considered. Here only one charge state is considered and other charge states are ignored. For ¹⁸¹Ta⁵⁹⁺, the reaction rate is 1.3 × 10⁻²/s for the electron beam with the energy of 0.33 keV, while for the beam of 17.721 keV, the reaction rate can be negligible due to the production rate being below the order of magnitude of 10^–30/s. Therefore, the total reaction rate of NEEC, i.e., R _NEEC = R _NEEC (E _er ) + R _NEEC (E _es ), is approximately equal to R _NEEC (E _er ).

The reaction rates of CE, i.e., R _CE (E _e ) = In _i σ _CE (E _e )L, I = j/e, are also considered, where CE cross sections were calculated using semi-classical theory [51]. The total reaction rates of CE, i.e., R _CE = R _CE (E _er ) + R _CE (E _es ) are listed in Table 2 too. For the 0.33 keV beam, the energy is too low and the reaction is forbidden; for the 17.721 keV beam, the CE rate and the NEEC rate are in the same order of 10^–2/s. This nucleus has a relatively high isomer generation rate R _tot = R _CE + R _NEEC in all the nuclei considered in this work. Indeed, some nuclei, such as ⁵⁷Fe, ¹¹⁹Sn, and ¹³⁷La, can be produced at rates greater than 0.1/s.

For other HCIs considered in this work, such as ¹⁵⁴Eu⁶³⁺, ¹⁵⁷Gd⁶²⁺, ²³⁷Np⁹¹⁺, ⁶³Ni²⁸⁺, ¹¹¹Cd⁴⁸⁺, ¹¹⁹Sn⁴⁸⁺, ¹³⁷La⁵³⁺, and ¹⁸⁷Re⁷⁵⁺, their NEEC excitation rates are much greater than the CE, which is very beneficial to prove the existence of the NEEC process. In particular, for ⁶³Ni²⁸⁺, ⁷³Ge²⁶⁺, ⁹⁹Ru⁴⁴⁺, ¹¹¹Cd⁴⁸⁺, and ¹²⁹I⁴⁷⁺ ions, the resonant and stripping electron beam energies are both less than the nuclear excitation energy. This means that the CE excitation is energetically forbidden, and only the NEEC excitation is present.

The decay of the isomers [52] are mainly through gamma transition and IC (collectively referred to as isomeric transition or IT decay); partly through beta decay such as β −, β +, orbital electron capture on nuclei; and a small amount through alpha decay, neutron emission, neutrino-nucleus processes, etc. However, all of our selected nuclei are stable, and their excited state decay is only IT decay.

The major backgrounds for NEEC searching come from RR and CE processes [50,53,54]. However, it should be noted that the NEEC is a resonance process, while CE processes are not resonance processes. This difference in resonance properties allows for the possibility of distinguishing the resonance peak from the flat background in the NEEC search. We have calculated the excitation rates of the possible CE and NEEC processes for different HCIs separately, as an example shown in Table 2. For most cases where both the electron beams are taken into account and the energies of the beams are smaller than 300 keV, cross sections due to CE are much smaller than the corresponding NEEC process cross sections.

The search for NEEC faces significant interference from the RR process, which is another major background. However, as a broad-resonant process, the RR process has anisotropic angular distributions [50], whereas the distribution of NEEC is isotropic. The combination of an isotropic distribution and a resonance peak can help distinguish the NEEC from the RR process. In addition, since the selected isomers are long-lived, they can be extracted from the ion trap to a clean environment for observation like [39,54] to avoid the interference of other background processes.

There may be competition between the NEEC and stripping processes. The former process involves adding an electron to a HCI, while the latter involves removing one. However, the time scale of nuclear excitation process is much shorter than the time scale required to strip electrons from outside the nucleus at high energies [55,56]. Therefore, the interference between the two processes can be negligible.

The expected thermal motion of ions in the trap is in the order of < 10 eV, which is smaller than the energy spread of the resonance electron beam at around 50 eV. As a result, the Doppler effect resulting from the ions’ thermal motion in the trap can be considered negligible [57]. Furthermore, other factors may cause changes in nuclear energy levels and their widths, such as the confinement magnet. The Zeeman effect causes energy levels of nuclei and atoms to split, with the energy splits (ΔE ∝B) typically being small due to limited confinement magnetic fields [58]. Therefore, given the electron beam’s energy spread of about 50 eV, the Zeeman effect can also be considered negligible.

Based on the estimated NEEC rates listed in Table 2 which are practically measurable and the controllable background noise signals discussed above, it is promising to search for the NEEC process using the proposed setup.

In summary, we propose an experimental scheme to study NEEC by utilizing an EBIT with two electron beams. The first electron beam strips atoms in the EBIT to high charge states, while the second electron beam matches the energy required for the NEEC process. Our calculations show that this approach yields high cross sections and sufficient reaction products. A resonance strength in the order of magnitude of 10^–3 b⋅eV is achievable by using nuclei like ¹⁸¹Ta⁵⁹⁺, ¹³⁷La⁵³⁺, ¹¹⁹Sn⁴⁸⁺, etc. The proposed scheme can be implemented with minimal modifications to existing EBIT setups, requiring only an additional electron gun and photon detectors. We are confident that our proposed scheme will stimulate additional research on NEEC processes and potentially confirm the existence of this phenomenon, which has been extensively investigated for over 50 years but has yet to be confirmed. Furthermore, we anticipate that our proposed scheme could be adapted to investigate other exotic processes (e.g., the NEET or EB process after further modifications) and potentially contribute to further advances in our understanding of nuclear physics.