State-Selective High-Energy Excitation of Nuclei by Resonant Positron Annihilation

In the annihilation of a positron with a bound atomic electron, the virtual gamma photon created may excite the atomic nucleus. We put forward this effect as a spectroscopic tool for an energy-selective excitation of nuclear transitions. This scheme can efficiently populate nuclear levels of arbitrary multipolarities in the MeV regime, including monopole transitions and giant resonances. In certain cases, it may have a higher cross sections than the conventionally used Coulomb excitation.

The lowest-order Feynman diagram of the NERPA process followed by nuclear de-excitation by emission of a photon. Thick lines denote nuclear states, double lines denote fermions in the Coulomb field of the nucleus, and wave lines represent real or virtual photons. tion is however not selective with respect to the nuclear energy levels.
In this Letter we show that NERPA has an attractive combination of the above advantages: the resonant character of the excitation and a significant cross section regardless of the multipolarity. We find that in certain cases its cross section can be comparable to or an order of magnitude higher than that of Coulomb excitation, leading to several applications. One of examples is the possibility to resonantly induce monopole transitions, that is particularly important for the studies of deformed nuclei. Another application is a collective nuclear excitation, the giant monopole resonance [17,18], also termed as "breathing" mode as it involves an oscillation of the nuclear volume, and provides the only way known for the experimental study of nuclear compressibility. In the case of giant nuclear resonances of any multipolarity, NERPA bears all the above-mentioned advantages of Coulomb excitation. A further property of NERPA is that experiments may be performed with neutral atoms, e.g. solid targets, without the necessity of stripping off the atomic electrons or accelerating the nuclei.
The NERPA process is the time reverse of e − e + internal pair conversion accompanying the decay of an excited nucleus. A concise model of that process is given in [35,36]. We extend the formalism of these works for the case of NERPA. Its transition rate (probability per unit time) is given by where j and j ′ are the total angular momentum quantum numbers of the e + and e − states, respectively, J i and J * f are angular momenta of the nuclear initial and final states, and the M -s and µ-s are the associated magnetic quantum numbers. Furthermore, α is the fine-structure constant, r n and r f denote nuclear and fermionic coordinates, and ω stands for the virtual γ energy. E and E ′ are the e + and e − energies, respectively, and j n and j f are nuclear and fermionic 4-currents [37]. Also, the notation [j 1 , ..., j n ] = n i=1 (2j i + 1) has been introduced. The further derivation involves a splitting of the double integral over nuclear and fermionic coordinates into a product of nuclear and fermionic parts. This can be performed in the non-penetration approximation, where the overlap integral of the bound-electron wave function within the nuclear volume is neglected. This simplification is generally used in the calculation of quantities related to the interaction of shell electrons with the nucleus [35,36,38,39]. Then it is possible to factorize the NERPA transition rate as P j ′ ,j N = β j ′ ,j N P γ , where P γ is the rate for the state reached by NERPA to decay by γ emission into the initial state.
For the coefficient β j ′ ,j N one may obtain expressions for any electric (λ = E) or magnetic (λ = M ) nuclear transition multipolarity, i.e. for any (non-zero) values of the angular momentum L of the virtual photon: where the radial part for the λ = E case is We have also introduced the following notation in terms of a 3-j are defined as in [35] with the analytical form of the radial Coulomb-Dirac wave functions for bound and free particles. All results are obtained here for the 1s 1/2 electron orbital, yielding the highest rate of NERPA. Fig. 2 shows the calculated dependence of the NERPA coefficient β j ′ ,j N for an electric-quadrupole (E2) nuclear transition between 1/2+ and 5/2+ states on the virtual photon energy ω in 115 Sn. Curves are given for two different possible positron angular momenta j = L ± 1/2; both values β j ′ ,L±1/2 N have to be taken into account in the calculation of the cross section. One can see that nuclear transitions about 2 MeV, corresponding to the maximum of these curves, are preferred. This behaviour is analogous to the energy dependence of the bound-free pair creation cross section in heavy ion collisions [41].
A promising approach to observe the NERPA process is to measure the photon emitted by the decaying nucleus, but it is a comparably weak effect superimposed on the background of other photon emission processes in the studied system. To circumvent this problem, as in Refs. [6,10], we consider nuclei with a long-living state, which can be populated by some radiative transition from the state excited by NERPA. The decay of this metastable state is the signature of the NERPA process, which may be measured with some time delay after the excitation takes place. The level scheme of the nucleus may be the one presented on Fig. 3 (a). Within this scheme, the state labeled as 2 is metastable, meaning that Γ 3 ≪ {Γ 1 , Γ 2 }, where the Γ-s denote transition widths as in Fig. 3(a). Γ N stands for the resonance width of the NERPA transition, which is related to the γ emission width as Γ N = P N = j [j]β 1/2,j N Γ 1 . The cross section of the total process in dependence on the positron energy E (and corresponding momentum p) is given by the expression (for a derivation of cross sections  Fig. 3 (a). Notations are as defined there and in the text. The symmetry ("symm.") of the nuclear levels involved -total angular momentum, parity -is given. Nuclear data are taken from Ref. [43]. 9.00 · 10 −4 1.39 · 10 −3 5.54 · 10 −4 Γ γ 2 (eV) 4.04 · 10 −4 9.9 · 10 −5 1.05 · 10 −4 Γ3 (eV) 3.965 · 10 −7 2.02 · 10 −10 4.06 · 10 −20 βN 7.6 · 10 −5 4.1 · 10 −5 1.9 · 10 −6 Γ nucl (eV) 1.83 · 10 −3 1.88 · 10 −3 6.6 · 10 −4 ΓK (eV) 33.0 7.9 7.3 σN(Eres) (b) 1.1 · 10 −4 1.5 · 10 −4 1.7 · 10 −5 σNγ(Eres) (b) 6.1 · 10 −6 3.1 · 10 −6 1.1 · 10 −6 of nuclear-resonant processes, see e.g. [38,39]): where E res is the resonance energy, Γ γ 2 is the width of the γ decay of the final NERPA state to the metastable state. This width is connected to the internal conversion (IC) coefficient α IC through Γ γ 2 = Γ 2 /(1+α IC ). Γ nucl = /τ is the total width of the considered nuclear level, expressed in terms of its lifetime τ . The total resonance line width is given as the sum of the nuclear and atomic K-shell widths [42]: Γ tot = Γ nucl + Γ K . The excitation cross section σ N can be obtained from the above formula by substituting the branching ratio Γ γ 2 /Γ nucl by 1. In Tab. I results for 3 different elements are presented, namely, 174 Hf, 115 Sn, and 115 In. All these elements possess the level scheme presented on Fig. 3 (a), and an E2 nuclear transition for the NERPA process. For 174 Hf, the cross section σ Nγ is the largest. However, here, the metastable level has a comparably short lifetime τ 2 , which may not be sufficient for an experimental differentiation of the signal from the direct bound-free e − e + photoannihilation process. The cross section and τ 2 for the case of 115 Sn and 115 In are most suitable for a possible measurement. The NERPA process is significantly enhanced for some elements such as e.g. 174 Hf and 115 Sn. Besides the increase of the β coefficient with Z, it is also because for these isotopes, the nuclear transition energy renders β nearly maximal (see Fig. 2). In the case of 115 In, it is rather the low positron resonance kinetic energy that boosts the total NERPA cross section due to the pre-factor 1/p 2 in Eq. (3).
We separately discuss the case of electric monopole (E0) nuclear transitions excited by the NERPA process. These 0 + → 0 + transitions are important for the investigation of the spectrum of deformed nuclei, since 0 + states in heavy deformed nuclei [44] are formed by a superposition of many oscillation modes and are objects of interest for microscopic nuclear models [45]. The direct one-photon nuclear excitation is forbidden in this case, therefore we compare the first (nuclear) step of the NERPA process to the dominant allowed nuclear decay channel of 0 + → 0 + nuclear transitions, namely, IC. The ratio δ j ′ ,j N of the probabilities of the NERPA process and IC in the non-penetration approximation can be written as P j ′ ,j N = δ j ′ ,j N P IC . Transition rates for the E0 case can be derived with the help of, e.g., Ref. [36].
E0 NERPA excitations enable the investigation of 0 + level cascades in heavy deformed nuclei. 168 Er is a typical example of such nuclei, having a 0 + ground state and a 0 + excited state at an energy of 1.217 MeV, which can decay by E2 γ emission to a metastable 2 + level at an energy of 79.8 keV with a lifetime of 1.853 ns (see Fig. 3 (b)). Here, the γ-decay of the metastable level follows the NERPA process. For the described level scheme in 168 Er, the lifetime of the excited 0 + level and the intensity of 0 + → 0 + transition are unknown, however, we can calculate the rate of IC transition between these levels, for instance following Ref. [46], and then, with the help of the δ coefficient derive the rate and width of the NERPA transition. Thus one can obtain the resonance cross section of the total NERPA-γ process through the reduced transition probability B E2 [47] of the 0 + → 2 + de-excitation, yielding σ Nγ (E res ) [b] = 1.12 · 10 −11 /(B E2 [e 2 fm 4 ]). The cross section may be determined experimentally, allowing one to obtain the reduced E2 probability for the second step of the NERPA process. This quantity is connected to the nuclear deformation parameter, thus the latter can be inferred from NERPA measurements and may be used to benchmark nuclear structure models.
We have identified the strongest NERPA excitation so far for a 13.5-MeV giant dipole resonance (GDR) in 208 Pb. Since for such resonances the nuclear level width is in the MeV range, these resonances also feature the highest integrated NERPA cross sections. For instance, the estimation of the NERPA cross section integrated with the effective resonance width reaches σ N (E)dE = 3.4 · 10 4 b·eV for the mentioned GDR in 208 Pb, exceeding previous values for e.g. 115 In by 8 orders of magnitude. GDRs can be efficiently excited even with a broadband positron source generated in laser-plasma interactions by existing novel methods [21,22]. E.g., for a positron beam with a 0.5-MeV width, Coulomb excitation of the GDR is not significant, since it requires positron kinetic energies higher by 2mc 2 than NERPA. Therefore, exciting GDRs with laser-generated positron beams may be a viable alternative to observe NERPA, circumventing the difficulties caused by the low nuclear line width in elements such as 115 In [8]. Furthermore, NERPA can be used for the excitation of a certain energy region of a giant resonance, enabling the investigation of the thermal evolution of the GDR width and structure. Normally such studies are done by Coulomb excitation [48,49] which does not allow a selective excitation but an energy-selective detection of subsequent γ decays. The problem of the theoretical description of the excitation of the certain part of the GDR is caused by the complicated resonance structure consisting of the superposition of the several nuclear states. To calculate the integrated excitation cross section in this case, one has to use some model for the GDR strength function. For instance, it can be the double-Lorentzian model of the strength function [50,51] for the case of the wide incoming positron distribution; or some more precise model, e.g. [52].
Another application of giant resonances where the energy selectivity of NERPA (due to its resonance structure) is beneficial is the feeding of highly deformed states by the low-energy GDR component [53]. This feeding effect is relevant for the investigation of the low-energy limit of the radiative dipole strength [54], which, in turn, has consequences in astrophysical simulations of the rprocess nucleosynthesis of exotic nuclei [55].
In summary, positron annihilation with a shell electron is put forward as an alternative means to induce nuclear transitions at the MeV level. Laser-produced positron beams may provide sufficient flux for efficient nuclear excitation. Cross sections are typically the largest in medium-Z and heavy elements with the nuclear transition energy near the maximum of the cross section about 2 MeV. This process is potentially relevant in numerous applications connected with positron-matter interaction, and is also anticipated to provide one with a novel means for the investigation of the structure of deformed nuclei and the energy spectrum of the giant resonances, as it can energy-selectively excite transitions of any multipolarity.
We acknowledge insightful conversations with Stanislav Tashenov and Christoph H. Keitel.