Simulation of nuclear isomer production in laser-induced plasma

With the development of laser technology, accompanied by increases in available laser intensity, a number of high-intensity laser (HIL)-induced nuclear physics phenomena have emerged.^1,2 Owing in particular to the use of the chirped pulse amplification (CPA) technique,³ the output power of lasers has steadily increased since the 1980s. Currently, laser intensities exceeding 10²³ W/cm² are achievable,^4–7 corresponding to electromagnetic field intensities E ≈ 10¹⁵ V/m and B ≈ 3 × 10⁶ T. The interaction of extreme electromagnetic waves with matter can directly or indirectly induce various nuclear processes. The rapid development of high-powered lasers has provided an unprecedented opportunity to study some important fundamental questions of nuclear physics under extreme conditions, as well as opening the way to possible new nuclear technology applications. Super-strong lasers can create high-energy-density environments, in which a variety of unique nuclear physics phenomena are attracting increasing attention,^8–10 including in particular, excitation and de-excitation of isomeric states.^11–17 

Controlling the excitation or de-excitation of nuclear isomers is a crucial issue in various fields, including nuclear batteries,^18,19 nuclear lasers,²⁰ nuclear clocks,^21–23 and nuclear astrophysics.^24–27 Nuclear batteries have very high energy densities. For example, ^178mHf, with a half-life of ∼31 years, is widely recognized as a suitable material for nuclear batteries.^18,19

Nuclear isomers are also expected to play important roles in enhancing the precision of time measurement. It is well known that the clocks with the highest precision currently available are atomic clocks,²² the working principle of which is based on spectral lines corresponding to atomic de-excitation, which are used as the time unit. The precision of these clocks depends on the spectral linewidths.²¹ With a deeper understanding of their excitation and de-excitation mechanisms, some nuclear isomers, such as ^229mTh and ^235mU, are expected to be used in nuclear clocks and provide the next generation of standard clocks.²² 

Furthermore, nuclear isomers are believed to play a crucial role during the process of cosmic nucleosynthesis.^24,27 In a high-temperature environment, energy levels of isomers and their surroundings will undergo significant changes with temperature, resulting in a greatly increased number of excitation and de-excitation pathways for the isomers, thereby affecting their effective lifetimes (half-lives). Such changes are highly sensitive to the environmental temperature. Therefore, an isomer in the usual sense is not necessarily the same as an isomer in astrophysics (with the latter being referred to in the literature as astrophysical isomers or “astromers”).²⁷ It has been predicted that the astromers may have significant astronomically observable effects on the rapid neutron capture process (r-process).²⁷ 

Nuclear isomers can typically be generated through processes such as heavy nuclear fission, collisions involving heavy ions, and interactions with HILs. Among these, the ultra-high density of laser plasmas stands out for its ability to induce ultrafast excitation of short-lived nuclear isomers, surpassing the abilities of the other two processes. Laser-driven plasmas provide an ideal environment for the study of nuclear isomer excitation or de-excitation. Advances in laser technology, coupled with growing understanding of plasma physics, have opened up new possibilities for investigating nuclear isomer-related phenomena, leading to a deeper understanding of nuclear physics.

In recent years, some theoretical and experimental progress has been made with laser-driven production of isomers. Pan et al.¹⁴ presented a feasibility study on photo-excitation production of four nuclear isomers (¹⁰³Rh, ^113m,115mIn, and ^176mLu) with an intense γ-ray source based on laser–electron Compton scattering (LCS). Feng et al.¹² reported the first experimental demonstration of femtosecond pumping of nuclear isomeric states (⁠$83m3$Kr) by the Coulomb excitation (CE) of ions with the quivering electrons induced by laser fields. Fan et al.¹⁷ experimentally demonstrated efficient production of ^93mMo via a ⁹³Nb(p, n) reaction induced by an intense laser pulse. These works are of significant value, providing the basis for further theoretical and experimental efforts in this field.

In this paper, we perform particle-in-cell (PIC) simulations of the laser-induced plasma dynamics using the Smilei code²⁸ and then investigate the nuclear isomer excitation processes in the plasma. In Sec. II, we provide a brief introduction to the theoretical background of HIL-induced nuclear excitation, including CE and nuclear excitation by electron capture (NEEC). In Sec. III, we present the details of the PIC simulations, and then discuss the results obtained. We present a summary of the study in Sec. IV. We will focus on the nuclear isomer production rates in a laser-induced plasma, as well as the possibilities for experimental confirmation of the NEEC mechanism.

Several mechanisms exist for exciting or de-exciting nuclear isomers, including direct photon excitation (PE), inelastic scattering (such as CE), nuclear excitation by electronic transition (NEET), NEEC, electron bridge (EB) processes,^29,30 photonuclear reactions (γ, Xn), and protonuclear reactions (p, Xn).^15–17 There are well-established theoretical and experimental results for the first three mechanisms.^30–32 However, despite considerable theoretical work, experimental evidence for the NEEC and EB mechanisms has yet to be obtained.^33–35 

The relative contributions of the various mechanisms in a HIL-induced plasma depend on the electron energy spectra and the specific structures of the target nuclei. Generally, as the maximum energy of the electron spectrum increases, so does the photon spectrum, owing to the bremsstrahlung effect. With increasing maximum photon energy, the possibility of the (γ, Xn) reaction producing an isomer also increases. Additionally, in cases where nuclei have specific energy levels, direct photon absorption may significantly contribute to isomeric excitation.

CE can excite isomeric states in HIL-induced plasmas. However, direct excitation through this mechanism may not be significant, owing to its relatively small cross-sections. An alternative path is through excitation of a nucleus from its ground state (g.s.) to a higher state (h.s.), followed by a decay back to the isomeric state (i.s.): g.s. → h.s. → i.s.

Although NEET can occur in HIL-induced plasma, cross-sections are again relatively small or even negligible.^31,32 In practice, the EB mechanism can be ignored, since it requires a narrowband laser,^32,36 and these are not widely available.

By contrast to the above mechanisms, the cross-sections for NEEC in HIL-induced plasmas are significant.^37,38 Although NEEC has been considered as an inverse process for internal conversion for almost 50 years,³⁹ it has yet to be experimentally confirmed.^33–35 However, theoretical calculations have indicated that NEEC should be a highly efficient mechanism for nuclear isomeric excitation,⁴⁰ and it may play a crucial role in astrophysical environments.^41–43 Therefore, much effort has been put into the search for evidence of the NEEC mechanism in HIL-induced plasmas.^12,13,38

In the following, we will briefly introduce the theories of CE and NEEC. Descriptions of the other mechanisms can be found in Refs. 29 and 30.

HIL can create high-energy-density environments, providing new methods for the production of short-lived nuclear isomers such as those of Ag and Ge, which have half-lives of the order of seconds and microseconds, respectively. For example, ⁷³Ge (natural abundance 7.75%) has an isomeric level at 13.3 keV (⁠$73m1$Ge) with a half-life of 2.91 µs. The two naturally occurring isotopes of silver, ¹⁰⁷Ag (abundance 51.35%) and ¹⁰⁹Ag (abundance 48.65%), have similar isomeric levels: $107m1$Ag at 93 keV, with a half-life of 44 s, and $109m1$Ag at 88 keV, with a half-life of 39 s. These two nuclides are almost identical, and so we will henceforth only discuss the isomers of ¹⁰⁷Ag.

In this work, we consider mainly ¹⁰⁷Ag and ⁷³Ge isotopes. Each of these has several possible transitions to isomeric states. Let us take excitation of the ¹⁰⁷Ag isotope as an example. As shown in Fig. 2, the transition could be excitation of the ground state (GS) to the third excited state (T03), followed by decay to the first excited state (T31), which will be denoted as GS → Third → First (T031). Excited levels above the third excited state (423.2 keV) are also possible. However, as discussed earlier, if the electron spectra are soft, such contributions are negligible, which is the case in the present study.

NEEC (Fig. 1) is a resonant process that involves the capture of free electrons into a bound atomic state. In this process, a portion of the electron’s energy is transferred to the nucleus as excitation energy.

We take both the CE and NEEC processes together to estimate nuclear isomeric products, employing Eqs. (3) and (6).

We use numerical simulations employing the PIC code Smilei²⁸ to investigate the nuclear isomer yield in laser-induced plasmas. Bonasera and co-workers^47,48 accelerated protons by target normal sheath acceleration (TNSA) using lasers and flat targets, achieving proton cutoff energies of around 35 MeV and proton conversion efficiencies of around 3%. Nanowire structures have been demonstrated to provide a powerful way to increase the deposition of laser energy into a target, enhancing laser absorption efficiency and the conversion efficiency of laser-generated hot electrons.^49–52 Therefore, we will employ a nanowire structure in this work.

In our model, the cylindrical Ag and Ge nanowires have a typical length of around L = 10.0 μm along the x direction, a typical diameter of around D = 0.2 μm, and a typical periodic spacing of around S = 0.45 μm. The wires are attached to a 2-μm-thick substrate of Ag or Ge foil (Fig. 3). The Ag nanowire target has an initial density of n₀ = 5.85 × 10²⁸ m⁻³. Here, $nc=mew02/4πe2$ is the critical density. The L, D, and S values of the nanowires have been set as realistic target parameters for experiments. The nanowire structures employed in this modeling could be fabricated using currently available nanotechnology techniques.⁵³ 

A p-polarized linear laser pulse is focused onto the front surface of the wires and is incident normally into the wire array target. The laser wavelength λ₀ = 800 nm, and the amplitude profile of the laser pulse is given by a(t) = a₀ exp[(t − t₀)/τ²], where a₀ = eE_L/m_ew₀c = 2 is the dimensionless amplitude of the laser electric field, t₀ = 25 fs, and τ = 10 fs is the laser pulse half-duration. E_L and w₀ are the peak electric field strength and the circular frequency of the laser, respectively. These values corresponds to a laser intensity of ∼8.6 × 10¹⁸ W/cm². Under this laser intensity and for a three-dimensional simulation box of 13 × 0.9 × 0.9 μm³, the laser energy is about 1.27 mJ.

We carry out a two-dimensional PIC simulation using a 2D box with dimensions 13 × 0.9 μm², sampled by 13 000 × 900 cells, which corresponds to steps of Δx = Δy = 10 Å. Absorbing boundary conditions are imposed for both the laser field and particles in the transverse direction. The boundary in the y direction is periodic, and the entire target is in a cold atomic state, with the “tunnel” model for field ionization being adopted.⁵⁴ Unless specified otherwise, the default parameters utilized in the simulation are those listed in Table I.

In simulating the interaction of a p-polarized laser with a nanowire array target, electrons and ions are first generated through the “tunnel” field ionization model, and electrons are pulled out of the nanowires. Through the direct laser acceleration (DLA) mechanism,^55,56 strong electric and magnetic fields from the laser pulses accelerate electrons and ions. The electron dynamics simulated by the PIC are illustrated in Figs. 4 and 5.

Figure 4 presents momentum spectra at different locations and times of electrons generated by the interaction of the laser with the Ag nanoarray target. It is notable that the directions of electron motion oscillate in response to the laser’s electromagnetic field. This oscillation increases the probability of electron–ion collisions, thereby facilitating the production of isomers through CE or NEEC.

Some nuclear excitation mechanisms, in particular NEEC in this study, depend strongly on ion charge states. In our simulation, we have adopted the internal field ionization model from the Smilei code,⁵⁴ which can be activated by defining ionization energies and an electron as the ionizing species. The ionization model assumes that the outermost electron always ionizes first.

As shown in Eq. (4), the NEEC cross-section is dependent on the charge state. Hence, it is crucial to consider the evolution of the charge states for both Ag and Ge nanowire arrays when evaluating the NEEC reaction rate in simulations. The corresponding results are shown in Fig. 6. These indicate the following. For the Ag nanowire array, the leading edge of the laser beam reaches the surface of the nanowires, and the charge state of the most abundant Ag ions gradually increases [Fig. 6(a)]. After about t > 70 fs, very few Ag ions have charge states ≤5, with Ag¹²⁺ dominating. There is a similar result for Ge ions, as shown in Fig. 6(b). The charge states around Ge¹⁰⁺ dominate.

The isomers produced by CE can be obtained by utilizing the output of the PIC simulation with the help of Eq. (3). Specifically, in the case of the CE mechanism, the cross-section of the T031 process depends strongly on the electron energy, as demonstrated in Fig. 7(a). For example, in the case of ¹⁰⁷Ag, the cross-section (magenta line) increases rapidly above the energy threshold for the reaction. The energy threshold E_th for the T031 reaction is the energy of the third level of ¹⁰⁷Ag.

On the other hand, the electron density exhibits an approximately exponential decrease with increasing energy according to n_e ∝ exp(−E_e/E₀₁) + exp(−E_e/E₀₂), where E₀₁ and E₀₂ are constants. In a plasma, the number density of high-energy electrons is much lower than that of low-energy electrons.

Equation (3) and the results in Fig. 7(a) suggest that the primary contribution to the final products stems from a relatively small range of energy. In the case of ¹⁰⁷Ag, this energy range is approximately [E_th, 3.5] MeV, which can be observed in Fig. 7(a). Similar results can be found for ⁷³Ge [Fig. 7(b)].

In this study, we have considered nanowire lengths L and substrate thicknesses d in the ranges L = 5–15 μm and d = 0–2 μm. However, in the simulations, the nanowire length is 10 μm and the substrate thickness is 2 μm, unless stated otherwise. Other parameters such as the wire diameter D = 200 nm and the distance between wires S = 450 nm have been kept constant.

Figure 8 shows the CE production rates of the isomers ^73mGe and ^107mAg. ^73mGe@T01 denotes the rate at which ⁷³Ge directly produces $73m1$Ge via CE from GS → First, and ^73mGe@T031 denotes the rate at which ⁷³Ge indirectly produces $73m1$Ge via CE from GS → Third → First. The first peak in Fig. 8, occurring at ∼35 fs, corresponds to the moment at which the Gaussian-shaped laser completely penetrates the nanowire array. The second peak, around 90 fs, marks the instant when the laser makes contact with the substrate of the array. There are additional peaks at 170, 270, and 380 fs, which appear to exhibit a periodic trend. These peaks may be related to the phenomenon of plasma wave echoing within the targets. The isomer $73m1$Ge (E₁ = 13.3 keV, T_1/2 = 2.91 µs) is produced via CE through the T01 and T031 processes at peak efficiencies of 6.3 × 10¹⁸ particles s⁻¹ J⁻¹ and 3.3 × 10¹⁸ particles s⁻¹ J⁻¹, respectively. The total rate of production of $73m1$Ge via CE is about 1.0 × 10¹⁹ particles s⁻¹ J⁻¹ through the T01 and T031 processes. The isomer $107m1$Ag (E₁ = 93.1 keV, T_1/2 = 44.3 s) is produced via CE through the T031 process with a peak efficiency of 2.95 × 10¹⁶ particles s⁻¹ J⁻¹.

As shown in Eq. (4), NEEC cross-sections depend on the charge state. Figure 9 shows the resonance strengths $Sq,αrNEEC≡∫σq,αrNEEC(Ee)dEe$ for both ¹⁰⁷Ag and ⁷³Ge. A notable observation is the strong dependence of the NEEC resonance strength on the capturing subshell. In the case of ¹⁰⁷Ag, $Sq,αrNEEC$ is relatively large, $≈10−11$ b eV, when the capturing subshell is L3(q = 41⁺) or K(q = 47⁺). For ⁷³Ge, the maximum $Sq,αrNEEC≈5×10−4$ b eV occurs when the capturing subshell is L3(q = 26⁺). Therefore, we disregard ¹⁰⁷Ag with its lower reaction cross-section and focus solely on calculating the NEEC reaction rate of ⁷³Ge.

Taking the PIC simulation results and using Eq. (6), we can calculate the production rates by NEEC. Figure 10 shows the production rates of the isomer ^73mGe by NEEC for ions in different charge states and at different times. ^73mGe (E₁ = 13.3 keV, T_1/2 = 2.91 µs) is produced at a peak efficiency of 1.65 × 10¹¹ particles s⁻¹ J⁻¹ via NEEC, compared with 1.0 × 10¹⁹ particles s⁻¹ J⁻¹ via CE. Therefore, distinguishing between production by NEEC and that by CE would be a very difficult task. However, considering that CE requires high-energy electrons, while NEEC involves much lower-energy electrons, the use of lower-intensity lasers to provide a large number of low-energy electrons and fewer high-energy electrons could potentially enhance the ratio r = N_NEEC/N_CE and thus create opportunities to observe the effects of NEEC.

We have conducted numerical simulations of nuclear isomer production by CE and NEEC in nanowire targets under femtosecond laser irradiation. In particular, we have focused on the production of isomers in the first excited states $73m1$Ge (E₁ = 13.3 keV, T_1/2 = 2.91 µs) and $107m1$Ag (E₁ = 93.1 keV, T_1/2 = 44.3 s). Our results reveal that for $73m1$Ge, CE contributes a peak efficiency of 1.0 × 10¹⁹ particles s⁻¹ J⁻¹, while, under the same setup, NEEC contributes a peak efficiency of 1.65 × 10¹¹ particles s⁻¹ J⁻¹. Despite the relatively low isomer production ratio between NEEC and CE, the laser-driven method still holds potential for confirming the existence of the NEEC mechanism, which has been studied for over 50 years, but has yet to be experimentally confirmed. The high efficiency of production by CE can be attributed to the high ion density and the high electron energy. The electrons are accelerated through nonlinear resonant interactions during the duration of the laser pulses. The combination of high efficiency, femtosecond timescale, and relatively easy production of short-lived nuclear isomers makes this approach highly favorable for investigating nuclear transition mechanisms and exploring the potential applications of nuclear γ-ray lasers.

This work is supported by the National Key Research and Development Program of China (NKPs) (Grant No. 2023YFA1606900) and the National Natural Science Foundation of China (NSFC) under Grant No. 12235003. The computations in this study were performed using the CFFF platform of Fudan University.

The authors have no conflicts to disclose.

Zhiguo Ma: Data curation (equal); Investigation (equal); Software (equal); Writing – original draft (equal). Yumiao Wang: Methodology (equal). Yi Yang: Methodology (equal). Youjing Wang: Software (equal). Kai Zhao: Software (equal). Yixin Li: Investigation (equal); Writing – review & editing (supporting). Changbo Fu: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Writing – original draft (equal); Writing – review & editing (equal). Wanbing He: Conceptualization (equal); Methodology (equal); Resources (equal). Yugang Ma: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Resources (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.