Nuclear Excitation by Electron Capture in Excited Ions
aa r X i v : . [ nu c l - t h ] F e b Nuclear Excitation by Electron Capture in Excited Ions
Simone Gargiulo, ∗ Ivan Madan, and Fabrizio Carbone † Institute of Physics (IPhys), Laboratory for Ultrafast Microscopy and Electron Scattering (LUMES),´Ecole Polytechnique F´ed´erale de Lausanne (EPFL), Lausanne 1015 CH, Switzerland
A nuclear excitation following the capture of an electron in an empty orbital has been recentlyobserved for the first time. So far, the evaluation of the cross section of the process has been carriedout widely using the assumption that the ion is in its electronic ground state prior to the capture.We show that by lifting this restriction new capture channels emerge resulting in a boost of variousorders of magnitude to the electron capture resonance strength. The present study also suggeststhe possibility to externally select the capture channels by means of vortex electron beams.
Isomers are potentially interesting for clean energystorage and release applications. The achievement ofa controlled and efficient extraction of the isomeric en-ergy has been a milestone for decades and is recently at-tracting growing attention [1–8]. In particular, recentlydemonstrated Nuclear Excitation by Electron Capture(NEEC) [9] has lot of potential in terms of control, asthe electron switch of the process can be manipulated bymeans of electron optics and wave function engineering[8, 10].NEEC is a process in which the capture of a free elec-tron by an ion results in the resonant excitation of a nu-cleus. The kinetic energy of the free electron, E r , needsto equal the difference between the nuclear transition en-ergy, E n , and the atomic binding energy released throughelectron capture, E b (i.e., E r = E n − E b ). The first iso-mer depletion induced by electron capture was recordedin a beam-based setup in 2018 [9]. The experiment re-mains fascinating as the strength of the detected signalis unexplained by state-of-the-art theory [11], presentinga discrepancy of about nine orders of magnitude.Until this work, the NEEC process has been consideredonly in ions which are in their electronic ground states(ground state assumption, GSA) [12–15], in ground stateions with a single inner-shell hole created by X-rays [16]or considering a statistical approach for electronic popu-lations in an average atom model [17–19]. In this letterwe examine the role of excited electronic configurationswithout any restrictions on the initial levels population.While the GSA allows for a straightforward account ofthe capture channels, it is too restrictive to unequivo-cally represent the real conditions taking place in out ofequilibrium scenarios. In fact, it has been shown that,for a given charge state q , the ground state configurationusually is not the most probable [20]. It is therefore im-portant to evaluate the cross sections of nuclear processesfor a wider range of electronic configurations.The GSA rules out the capture in the innermost shellsfor partially filled ions. For example, one can have K-capture till two electrons fill the 1s orbital. However,even for fully ionized nuclei, NEEC into K-shell may ∗ simone.gargiulo@epfl.ch † fabrizio.carbone@epfl.ch be forbidden if the energy released through a K-capture( E Kb ) exceeds the nuclear transition energy (i.e., E r < Ge.In Fig. 1 we compare both the conventional and ourapproach. In Fig. 1a NEEC takes place in an ion underthe GSA. A variant of NEEC — i.e., NEEC followed bya fast x-ray emission (NEECX) — considers the captureof the electron in a higher energy electronic shell whilethe ion is still in its electronic ground state, a situationin which the GSA still holds [21, 22], see Fig. 1b. In-stead, Fig. 1c represents the case in which the GSA doesnot hold: here, NEEC can occur even in excited ions(NEEC-EXI) and the consequences of such a scenarioare discussed below. ba NEEC – GSA NEEC – EXINEECX – GSA
Atomic levelsNuclear levels c FIG. 1. Atomic configurations in case of electron capture:conventionally, the ion is considered to be in its nuclear andelectronic ground states, while the capture either leaves theion in the electronic ground state (a) referred to as NEEC,or bring it in an electronic excited state (b), referred to asNEECX. In (c), electrons can be distributed all over K, Land M shells. Γ represents the width of the atomic (Γ At nl ) andnuclear (Γ N ) transitions. The NEEC process can be described as the formationof an excited compound nucleus, in which the entrancechannel is represented by the incoming electron. As de-rived in the Supplemental Material [23], the integratedNEEC cross section, called resonance strength S NEEC ,can be expressed as [14, 24–27]: S q,α r NEEC = Z σ q,α r NEEC ( E ) dE = S λ α q,α r IC Γ γ , (1)where S = (2 J E +1)(2 j c +1) / (2 J G +1)(2 j f +1) , λ e is the electronwavelength and Γ γ is the width of the electromagneticnuclear transition. J E , J G and j c and j f are the nu-clear spins of the excited and ground states and the to-tal angular momenta of the captured and free electrons,respectively [26]. α q,α r IC is the partial internal conversioncoefficient (ICC) that depends on the final electronic con-figuration ( α r ) and on the ion charge state q prior to theelectron capture. Under the GSA, the initial electronicconfiguration ( α ) is uniquely defined by the charge state q and the number of available channels for capture in aparticular subshell nl j is strongly limited. By contrast,in NEEC-EXI the coefficient α q,α r IC also depends on α ,thus it has to be expressed as α q,α ,α r IC .ICCs for neutral atoms are computed using the frozenorbital (FO) approximation based on the Dirac-Fock cal-culations [28], whereas for ionized atoms a linear scalingdependence is assumed to obtain an order of magnitudeestimate for S NEEC [15, 29]: α q,α ,α r IC E q,α ,α r b = α q =0 ,nl j IC E q =0 ,nl j b n h n max ! , (2)where E q,α ,α r b and E q =0 ,nl j b are the binding energies forions in the charge state q and neutral atoms, respectively.Their ratio accounts for the increase of the ICCs with theionization level [17, 29]. The ratio between the present n h and the maximum n max number of holes in the capturesubshell nl j accounts for the decrease of the ICCs forpartially filled subshells [29]. The binding energies forneutral atoms are taken from tables [30, 31], while theones for highly ionized atoms are calculated with FAC[32], obtained as energy difference between the initial( α ) and final electronic configurations ( α r ). Accuracyof these levels is assessed to be in the order of few eV[33]. When approaching the threshold E r ≤ q , electrons areassigned to a particular shell from the innermost to theoutermost (K, L, M) encompassing all possible combina-tions. All these states are used as initial configurations α . In case the electron involved in the capture breaks theorbital angular momentum coupling in the initial atomicconfiguration α , the expression of the NEEC resonancestrength in Eq. 1 is furtherly complicated by an addi-tional coefficient Λ, expressing the recoupling probabil-ity between the initial ( α ) and final electronic configu-rations ( α r ) [35–40]: S q,α ,α r NEEC = Λ
S λ α q,α ,α r IC Γ γ . (3) In this letter, the recoupling schemes for ions with up tofour electrons filling the orbitals have been considered.Further details about the expression of Λ and electronrecoupling are given in the Supplemental Material [23].Besides the energy matching, also the angular momen-tum must be conserved in the NEEC process. In par-ticular, the atomic total angular momentum change be-fore and after the capture must match the nuclear spinvariation. Thus, once the allowed channels are identifiedthrough the selection rules of the particular recouplingunder study, the individual total angular momentum j f of the free electron can be evaluated as [41, 42]: j f = (cid:12)(cid:12) ∆ J N − ∆ J At (cid:12)(cid:12) , (4)where ∆ J N represents the nuclear spin change and ∆ J At the variation of the atomic total angular momentum.Since ICC predictions for neutral atoms have beenshown to have less than 1% uncertainty level when com-pared to experimental data [28, 43, 44], the main approx-imation of the presented approach resides on the evalu-ation of ICCs for ionized atoms, expressed in Eq. 2.To test the method, it can be applied under the GSAand compared with the more advanced theory presentedin Ref. 45, based on Feshbach projection operator for-malism, for Fe. The computed resonance strengths for Fe, using Eq. 1, considering the nuclear excitation be-tween the stable 1 / − ground state and its first 3 / − excited level at 14 . S NEEC value obtained for thecapture in the 1s / shell of Fe , 1 . × − b eV,is in good agreement with the value of 1 . × − b eVobtained from Ref. 45, once the contribution of the sub-sequent γ -decay of the excited nucleus is removed.Applying the GSA to the Ge nuclear transition of E n = 13 . / + ground and the5 / + first excited states provides 47 L- and M-channelsfor q = [29+ , E Kb becomes smaller than E n , NEECinto the K shell is possible. For Ge this condition ismet for q = 29+, for which 100 K-capture channels havebeen unveiled, as shown in Fig. 2b. Most of these K-channels (78) are characterized by an initial electronicconfiguration α of the type 1s nl j nl j and occurin the energy range E r = [0 , . α is 1s nl j and E r = [48 . , . α = { nl j } are still forbidden, since E Kb is larger than E n by about 200 eV at q = 29+. Resonance strengths forhigher charge states are shown in the Supplemental Ma-terial [23]. Notably, the higher number of channels iden- a -6 -5 -4 -3 -2 -1 NEEC – GSA
K-channels forbidden L-channels <> qq << M-channels q ÷ -2 NEEC – EXI j f b K-channels L-channels M-channels << C ap t u r e s he ll -4 -3 -1 q FIG. 2. (a) Resonance strengths for capture in the L- (green box) and M-shell (blue box) in case of Ge with q = [29+ , E Kb > E n ). For q = 29+ E Kb < E n , however the K-shell is completely filledand capture can not occur (insets). (b) Resonance strengths for Ge in case all the possible combinations of initial andfinal electronic configurations are taken into account, for q = 29+. Each resonant channel is represented by a solid line, withits colors indicating the capture orbital, while the color of the circled marker represents the value of j f . L- and M-channelsare partially displayed to improve readability, while all the K-channels are shown. Both graphs share the same y-scale andcolorbars. The horizontal green and magenta lines indicate the highest S NEEC , under GSA, for L- and M-shells — occurringat [ q, α r ] = [32+ , / ] and [ q, α r ] = [32+ , / ] — respectively. tified in NEEC-EXI is not only due to the several initialconfigurations considered, but also to the increase of thecapture channels available for a single excited configura-tion α compared to the ground state counterpart. Thereason is that excited configurations can have a largernumber of open shells, thus the number of final configu-rations that can be generated are generally more numer-ous due to the higher number of combinations possiblefor the electron couplings.Fig. 2 compares the resonance strengths of the newlyopened K-channels and L- and M-channels for NEEC-GSA and NEEC-EXI. Here, only shells up to the M havebeen considered, since α with electrons in higher shellsdo not provide sufficient screening for a K-capture at q = 29+. Selected channels are reported in Table I and,when possible, are compared with those evaluated withthe GSA procedure for which Eq. 3 reduces to Eq. 1,since Λ = 1, and results coincide. It is worth to mentionthat the maximum value obtainable for the resonancestrength with and without GSA differs by more than twoorders of magnitude in the interval q = [29+ , S NEEC in the L- and M-shells instead are comparablebetween the two cases.There are two main factors defining the final S NEEC value for a given character of the nuclear transition and E n : (i) the resonance energy of the capture channel and(ii) the value of the partial internal conversion coefficient α q,α ,α r IC . (i) Because of the resonant nature of the NEECprocess, S NEEC increases dramatically when the energyreleased through electron capture nearly matches the nu-clear transition. (ii) α q,α ,α r IC depends on the overlap be- tween the electron and the nucleus wave functions. Inthe case of Ge, the enhancement found for the K-shell,compared to the highest value obtained under the GSAoccurring for an L3 subshell, is solely due to an increaseof the electron wavelength, since α q, KIC < α q, L3IC .It is thus important to comment on the accuracy ofthe calculated energy levels. In the Supplemental Ma-terial [23], we compare the 38 energy levels available forGe, obtained from the NIST website [46], and the samereproduced by FAC. The results show a good agreementwith discrepancies between these levels usually smallerthan 1 eV and in all cases comparable with the accuracyreported for the E2 nuclear transition of Ge. Altoughtthe S NEEC values of the nearly-resonant energy levelsare affected by the accuracy of FAC, 27 K-channels arepresent in the range E r = [0 ,
10 ] eV and 18 still forbid-den in the range E r = [ − , Tcand
Te. In the latter case, contrary to what happensfor Ge, a further increase of S NEEC is expected dueto a higher value of α q=0 , KIC compared to α q=0 , L i IC , with i ∈ { , , } .An increase of the resonance strength is particularlyvaluable when NEEC is compared to competitive pro-cesses, such as the direct-photoexcitation (DP) in thelaser-generated plasma scenario [14, 17, 19, 47]. Here,the discrimination of the two processes relies on the to-tal number of excited nuclei, proportional to the pho-ton/electron flux in plasma and corresponding resonancestrengths. In table-top laser based setup, the photon fluxcan exceed the electron flux by several orders of magni- TABLE I. Resonance strengths for Ge in case of NEEC-EXI. For a given final electronic state ( α r ) all the relative parentconfigurations α , that through electron capture can lead to it, are taken into consideration. The OAM of the free electron j f and the resonance energies, intended as E r = E n − E b , are reported. When possible, a comparison with the conventionalderivation (GSA) is also presented. In blue the subshell nl j in which the capture occurs. Ge NEEC-GSA NEEC-EXI q Initial Configuration ( α ) Final Configuration ( α r ) j f E r [eV] S NEEC [b · eV] S NEEC [b · eV]32+ − / / . × . × − . × −
31+ 1s / . × . × − . × −
31+ 3d / / / / . × Not Allowed 2 . × −
30+ 1s / . × . × − . × −
30+ 3d / / / / / / . × Not Allowed 4 . × −
29+ 1s / / . × . × − . × −
29+ 3p / / / / / / / / . × Not Allowed 1 . × −
29+ 3p / / / / / / / / . × Not Allowed 8 . × −
29+ 1s / / / / / .
62 Not Allowed 7 . × −
29+ 1s / .
87 Not Allowed 5 . × −
29+ 1s / / / .
32 Not Allowed 3 . × −
29+ 1s / / / .
15 Not Allowed 2 . × −
29+ 1s / / / / / .
39 Not Allowed 2 . × −
29+ 1s / / / .
18 Not Allowed 3 . × − tude [14, 23, 48, 49]. This might hinder the observa-tion of NEEC even for such promising nuclei as Ge,for which the DP resonance strength of the E transi-tion is S γ = 1 . × − b eV, significantly smaller thanthe highest S NEEC = 1 . × − b eV, obtained underthe GSA. Conversely, lifting the GSA allows for the ap-pearance of capture channels in the K-shell characterizedby higher S NEEC values. This is particularly relevant ifan additional external electron source is considered. Fora few keV temperature plasma, the flux of electrons atlow energies corresponding to K-channels is small and ofthe order of 10 cm − s − eV − . Under this condition,the use of an external adjustable electron source [23, 50]could allow to overcome the deficit in the electron fluxand decouple it from other plasma parameters.In out-of-equilibrium scenarios, excited electronic con-figurations might be more likely to occur [20] and thesame can hold true for the Mo isomer depletion ofRef. 9. Under the GSA, despite the NEEC excitationprobability P αq is the highest for capture in the L-shell,the corresponding total NEEC probability P is negligi-ble with respect to the contributions coming from M-, N-and O-shells [11]. This happens because the ion fractionin the charge state q ≥
33+ required for L-shell vacan-cies is extremely small. Under NEEC-EXI, although theK-capture remains strictly forbidden, the presence of va-cancies in the L-shell at the resonance condition — evenfor q <
33+ — could make NEEC excitation probability P αq for L-channels no longer insignificant.Till now, in NEEC scenarios only the energy matchingbetween free electrons, bound states and nuclear tran-sitions has been addressed. We point out that angularmomentum matching, expressed by Eq. 4, suggests thepossibility to select the capture channel by tuning theindividual orbital angular momentum (OAM) l ~ of anexternal free electron beam [51, 52], using phase platesor chiral plasmons [10, 53–55]. Fig. 2 also highlights awider range of j f available in case of NEEC-EXI. Whenusing such OAM-carrying electrons (called vortex beam)the principle of detailed balance used to derive Eq. 1does not hold in the same form [23, 27, 56], thus leadingto different values for the S NEEC . This could also reveala way to enhance the electron capture in the innermostshells. The presence of this additional degree of freedom,compared to the solely free electron energy, could open tothe possibility of an external control of the NEEC rate ina plasma scenario: by providing pulsed vortex electronsat the resonant energy, the isomer depletion rate can becontrolled by dynamically varying the OAM of the vortexbeam [54, 57, 58].In conclusion, we have shown that the common as-sumption that NEEC takes place in an ion in its elec-tronic ground state significantly restricts the availablechannels. By lifting this condition, we have shown that in Ge the NEEC resonance strengths gain several ordersof magnitude. Thus, this work heralds the possibility ofa re-evaluation of the isotopes prematurely disregardedand those already in use in out-of equilibrium scenarios.These findings could open a new route for an externally-controlled nuclear excitation by providing resonant elec-trons with engineered wavefunctions, thus selecting thepromising channels for the on-demand isomer depletion.
ACKNOWLEDGMENTS
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Simone Gargiulo, ∗ Ivan Madan, and Fabrizio Carbone † Institute of Physics (IPhys), Laboratory for Ultrafast Microscopy and Electron Scattering (LUMES),´Ecole Polytechnique F´ed´erale de Lausanne (EPFL), Lausanne 1015 CH, Switzerland
CONTENTS
NEEC resonance strength derivation 1Recoupling coefficients 2NEEC resonance strength for Fe and Ge under Ground state assumption 6Accuracy of the energy levels computed with the Flexible Atomic Code (FAC) 8NEEC in Excited Ge ions 9NEEC in laser-plasma scenario with GSA 10References 11
NEEC RESONANCE STRENGTH DERIVATION
Conventionally, the NEEC process can be described as the formation of an excited compound nucleus, in which theentrance channel α is represented by the incoming electron. Thus, it is possible to write the NEEC cross section asthe Breit-Wigner one-level formula [1–3]: σ NEEC = πk Γ α Γ r ( E − E r ) + Γ / = 2 π k Γ α L r ( E − E r ) , (1)where k e is the electron wave-number, Γ r is the natural resonance width given by the sum of the atomic (Γ At ) andnuclear (Γ N ) widths, Γ α represents the transition width of the entrance channel α and L r is a Lorentzian functioncentered at the resonance energy of the free electron E r .Γ α is defined by the microscopic NEEC reaction rate Y NEEC [4] via Γ α = ~ Y NEEC , that is related to the internalconversion rate ( A IC ) through the principle of detailed balance [5]. Thus, in case of unpolarized beams we have: Y NEEC = (2 J E + 1)(2 j c + 1)(2 J G + 1)(2 j f + 1) A IC (2)where J E , J G and j c and j f represent the nuclear spins of the excited and ground states and the total angular momentaof the captured and free electrons, respectively [3]. The NEEC cross section is then given by: σ NEEC = (2 J E + 1)(2 j c + 1)(2 J G + 1)(2 j f + 1) λ IC L r ( E − E r ) , (3)where λ e is the electron wavelength. More precisely, the microscopic NEEC and IC rate depend on the particularsubshell nl j in which the electron is captured and charge state q of the ion prior the electron capture. This informationis condensed in the evaluation of the partial internal conversion coefficient α q,α r IC , that depends on the final electronicconfiguration ( α r ) and on the ion charge state q prior to the electron capture. Under the GSA, the initial electronicconfiguration ( α ) is uniquely defined by the charge state q — i.e. the electronic ground state — while the finalelectronic configuration ( α r ) depends also on the particular capture channel. Thus, Eq. (3) can be expressed as: σ q,α r NEEC = S λ α q,α r IC Γ γ L r ( E − E r ) , (4)where S = (2 J E +1)(2 j c +1) / (2 J G +1)(2 j f +1) and Γ γ is the width of the electromagnetic nuclear transition. ICCs for neutralatoms are estimated by using the frozen orbital (FO) approximation based on the Dirac-Fock calculations [6], whilefor ionized atoms a linear scaling dependence is assumed [7, 8]: α q,α r IC E q,α r b = α q =0 ,nl j IC E q =0 ,nl j b n h n max ! , (5)where E q,α ,α r b and E q =0 ,nl j b 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 [8, 9]. 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 partiallyfilled subshells [8]. The binding energies for neutral atoms were taken from tables [10], while the ones for highlyionized atoms are calculated with FAC [11], obtained as energy difference between the initial ( α ) and final electronicconfigurations ( α r ). The resonance energy E r is then obtained as E n − E b for the specific channel considered. Theintegrated NEEC cross section, called resonance strength S NEEC , is then defined as: S q,α r NEEC = Z σ q,α r NEEC ( E ) dE = S λ α q,α r IC Γ γ . (6)As described in the main text, when NEEC is occurring in excited ions, the coefficient α q,α r IC also depends on theparticular initial electronic configuration α and it has to be expressed as α q,α ,α r IC . As a consequence, also the NEECcross-section and the resonance strength have to be represented as σ q,α ,α r NEEC and S q,α ,α r NEEC , respectively.
RECOUPLING COEFFICIENTS
In NEEC-EXI, for a given charge state q , electrons are assigned to a particular shell from the innermost to theoutermost (K, L, M) encompassing all possible combinations (e.g., for 3 electrons all the cases between 1 s s and3 d / are considered; for 4 electrons all the cases between 1 s s and 3 d / ). All these electronic states are used asinitial configurations α .When considering the electron capture in this context, it has to be taken into account the possible recouplingbetween the electron involved in the capture and those that are already in the atom. The case of initially fully ionizedatom is trivial since no recoupling is occurring. In case the capture leads to the formation of an ion having twoelectrons, the selection rules are satisfied if the spectator electron (not involved in NEEC) preserves its orbital angularmomentum during the process. The NEEC cross-section would be non-zero only if this condition is satisfied. For thecase of three electrons, we can recognize two situations: (i) the capture does not break the coupling or (ii) the capturebreaks the coupling. With j , j and j c we denote the total angular momenta of the two spectator electrons and theone involved in the capture, respectively. Case (i) occurs when [12]:1. j and j firstly couple to J ,2. J then couples with j c forming J ,where J is the initial angular momentum, not broken by the capture. The NEEC cross-section can be considerednon-zero only when the orbital angular momenta of the two spectator electrons and their coupling (thus j , j , J )remain unchanged among the initial and final states. The other possibility is (ii):1. j and j c firstly couple to J ,2. J then couples with j forming J .Here, the capture breaks the initial coupling and the expression of the resonance strength in Eq. 6 has an additionalcoefficient [12–14]:Λ = |h [ j , ( j , j c ) J ]; J | [( j , j ) J , j c ]; J i| = (2 · J + 1)(2 · J + 1) (cid:26) j j J j c J J (cid:27) , (7)where Λ expresses the probability that a system with a coupling scheme defined by the bra vector h [ j , ( j , j c ) J ]; J | will be found in the scheme | [( j , j ) J , j c ]; J i [14]. If a peculiar coupling is possible or not depends on the Wigner6j-symbol. Notice that here J also represents the total orbital angular momentum of the two-electron atomic system before the electron capture. With three electrons, three nontrivial coupling schemes exist and as soon as more electrons( n ) are added to the ion, the number of possible couplings increases as (2 n − α having a charge state q = ( Z − • / / → / / ; • / → / / ; • / / → / / ; • / → / / .The recoupling coefficients associated with the first three cases are: h [( j , j ) J , j ] J , j ; J | [( j , j ) J , j ] J , j ; J i = ( − θ R (cid:26) j j J j J J (cid:27) (cid:26) j J J j J J (cid:27) , (8) R = p (2 · J + 1)(2 · J + 1)(2 · J + 1)(2 · J + 1) , h [( j , j ) J , j ] J , j ; J | [( j , j ) J , j ] J , j ; J i = ( − θ R (cid:26) j j J j J J (cid:27) (cid:26) j J J j J J (cid:27) , (9) R = p (2 · J + 1)(2 · J + 1)(2 · J + 1)(2 · J + 1) , h [( j , j ) J , j ] J , j ; J | [( j , j ) J , j ] J , j ; J i = ( − θ R (cid:26) j J J j J J (cid:27) , (10) R = p (2 · J + 1)(2 · J + 1) , respectively. In these equations, the phase factor — indicated by θ i — is not reported since it is irrelevant inthe evaluation of the probability Λ = |h a | b i| , with a and b being the two state vectors expressing the coupling.Differently, the selection rule for the capture in the outermost shell requires that the electrons not involved in NEECconserve their individual j i and their intermediate couplings J ik , J ikl .These recoupling coefficients, reported in Eqs. 8-10, can be understood by graphical means using Yutsis notation[13] and binary trees [14, 15], presented in Fig. 1. Each pair of binary trees, whose leaves are labelled with the fouruncoupled angular momenta, can be connected by two types of elementary operations: exchange and flop [14, 16, 17].An exchange , represented by a dashed arrow, does not lead to a rearrangement of the orbital angular momenta, butto a swap of the j i around one node. Thus, the relative transformation coefficient corresponds to a phase factor.The flop operation instead, shown as a solid arrow, is effectively a recoupling relating two trees with two alternativenets connecting the leaves. This latter transformation is defined by a Racah coefficient, proportional to a (represented by the rhomboidal Yutsis graph in Fig. 1).Electron capture can occur in many other circumstances. As for example, we can have an initial configuration α with q = ( Z − • / / → / / ; • / / → / / / ; • / / → / / .In Eqs. 11-13 are reported the associated recoupling coefficients. FIG. 1. Binary trees connecting the three coupling schemes given in Eqs. 8-10. Starting from one state, applying a sequenceof the elementary operations presented, it is possible to arrive at the desired final state.
Exchange and flop operations arerepresented by green dashed and blue solid lines, respectively. The rhombuses graphically represent, in Yutsis notation [13],the Wigner 6-j symbols for the considered flop operation. Eq. 8, for example, can be obtained following the path going fromthe state | [( j , j ) j ] j i , boxed in purple, to the state vector | [( j , j ) j ] j i , indicated by a black square, as multiplication oftwo Wigner 6-j symbols (with the relative square root terms R i due to the two flop operations) and an additional phase factorcoming from the exchange operation. Eq. 9 is given by the path going from the cyan to the black boxes, while Eq. 10 is thepath connecting the red and black boxes. h [( j , j ) J , ( j , j ) J ]; J | [ j , ( j , j ) J ] J , j ; J i = ( − θ R (cid:26) j j J J J J (cid:27) , (11) R = p (2 · J + 1)(2 · J + 1) , h [( j , j ) J , ( j , j ) J ]; J | [ j , ( j , j ) J ] J , j ; J i = ( − θ R (cid:26) j j J J J J (cid:27) , (12) R = p (2 · J + 1)(2 · J + 1) , h j , [( j , j ) J , j ] J ; J | [ j , ( j , j ) J ] J , j ; J i = ( − θ R (cid:26) j J J j J J (cid:27) , (13) R = p (2 · J + 1)(2 · J + 1) , Another relevant case is given by the initial electronic configuration ( α ) of the type 1s l l with q = ( Z − α r ) resulting from a capture in the K-shell would lead to the following scenario: • / / → / / .This represents one example of the 100 K-capture channels identified in the manuscript. The associated recouplingcoefficient is: h [( j , j ) J , j ] J , j ; J | [( j , j ) J , j ] J , j ; J i = ( − θ R (cid:26) j j ✟✟ J
12 0 j J J (cid:27) (cid:26) j J J j J J (cid:27) , (14) R = p (2 · J + 1)(2 · J + 1)(2 · J + 1)(2 · J + 1) = p (2 · J + 1)(2 · J + 1)(2 · J + 1) . NEEC RESONANCE STRENGTH FOR FE AND GE UNDER GROUND STATE ASSUMPTION
Considering the framework here presented, Eq. 6 leads to the resonance strengths shown in Fig. 2 and in TableI for Fe; in Fig. 3 and in Table II for Ge. For both isotopes, the only capture in the K- and L-shells have beenconsidered, while the charge state varied between the one of the bare nucleus ( q = Z +) all the way to the closure ofthe K- and L-shells ( q = ( Z − Fe only 2 K-capturechannels exists, versus 27 L-channels in the interval q = [17+ , Genuclear transition reveals a difference: Ge only has the 27 channels for the L-shell for q = [23+ , j f -6 -5 -4 -3 -2 -1 C ap t u r e s he ll NEEC – GSA
FIG. 2. Resonance strengths for the K- and L-shell channels of Fe, under the GSA. Charge state q ranges between 26+ (barenucleus) to q = 17+ (after the electron capture the K- and L-shells are both closed).TABLE I. NEEC resonance strength S NEEC , partial internal conversion coefficient α q,α r IC and the energy of the continuumelectron E r for various charge state q (prior the electron capture) and capture shells α r , in case of Fe. Fe has its first excitedstate at 14 . α r is used to indicate the capture channel, as theyare uniquely connected once the charge state q is assigned. q α r E r [keV] α q,α r IC S NEEC [b · eV]26+ 1s / .
14 9 .
98 1 . × −
26+ 2s / .
09 1 .
99 1 . × −
26+ 2p / .
09 0 .
14 8 . × −
26+ 2p / .
11 0 .
06 7 . × −
25+ 1s / .
59 4 .
75 6 . × −
24+ 2s / .
37 1 .
75 1 . × −
24+ 2p / .
42 0 .
12 7 . × −
20+ 2p / .
84 0 .
04 4 . × −
17+ 2p / .
15 0 .
008 9 . × − j f C ap t u r e s he ll NEEC – GSA -6 -5 -4 -3 -2 -1 FIG. 3. Resonance strengths for the L-shell channels of Ge, under the GSA. Charge state q ranges between 32+ (bare nucleus)to q = 23+ (after the electron capture the K and L shells are both closed).TABLE II. NEEC resonance strength S NEEC , partial internal conversion coefficient α q,α r IC and the energy of the continuumelectron E r for various charge state q (before the capture) and capture shells α r , in case of Ge. Ge has its first excited stateat 13 . α r is used to indicate the capture channel, as they areuniquely connected once the charge state q is assigned. q α r E r [keV] α q,α r IC S NEEC [b · eV]32+ 2s / .
74 72 .
02 2 . × −
32+ 2p / .
74 655 .
55 2 . × −
32+ 2p / .
79 1167 .
65 1 . × −
31+ 2p / .
95 617 .
59 2 . × −
31+ 2p / .
99 1101 .
94 1 . × −
30+ 2s / .
09 64 .
93 2 . × −
30+ 2p / .
15 579 .
29 1 . × −
29+ 2p / .
30 996 .
35 1 . × −
26+ 2p / .
71 859 .
36 1 . × −
25+ 2p / .
84 611 .
68 3 . × −
23+ 2p / .
10 181 .
81 2 . × − ACCURACY OF THE ENERGY LEVELS COMPUTED WITH THE FLEXIBLE ATOMIC CODE (FAC)
For the four ions of Ge, i.e. q = [29+ , Ge XXXII Ge XXXIGe XXX Ge XXIX
FIG. 4. Energy differences between the levels computed with FAC and the same energy levels available from the NIST database[18] for the charge state interval q = [29+ , Agreement is excellent for the Hydrogen-like ion Ge XXXII, where the standard deviation of the energy differences isof 6 . . .
32 eV, while their standard deviation is of 0 .
44 eV. The largestdiscrepancy of − .
38 eV is for the 1s ( S ) level. 14 of these NIST levels are extrapolated or interpolated startingfrom the two known experimental values. A good agreement, with a standard deviation of 0 .
78 eV and an averagevalue of the energy differences of − .
08 eV, is found also for the Ge XXX ion. Apart from the 1s ( D / ) level,all the energy differences are < D / level reports a discrepancy of − .
48 eV. The 7 energy levels reportedon the NIST database for this ion are experimentally observed, while the ionization energy is theoretically predicted.For Ge XXIX, only 4 suitable levels are present on the NIST database. The mean value of the energy difference is of0 .
67 eV, while the standard deviation is of 0 .
73 eV, mainly given by the discrepancy observed for the 1s ( P )level, that is of 1 .
68 eV. Here, the 3 electronic excited states are experimentally observed, while the ionization energyis obtained from extrapolation.The quality of the FAC calculations can be appreciated when discrepancies here reported are compared with theenergy spread over which the studied electronic configurations persist, i.e. of few keV.
NEEC IN EXCITED GE IONS
As described in the main text, lifting the GSA provides a total of 32823 capture channels in the charge state interval q = [29+ , E Kb ), till it becomes smallerthan E n , and NEEC into the K shell starts to be possible. In particular, for Ge this condition is met for q ≤ -4 -6 -5 -4 -2 -3 -2 -1 C ap t u r e s he ll f FIG. 5. Resonance strengths for Ge with q = [29+ , α ) and final ( α r )electronic configurations are taken into account, as a function of the charge state q and of the resonance energy E r . Eachresonant channel is represented by a solid line, with its colors indicating the capture orbital, ranging from the K to the M shell,while the color of the circled markers represent the value of j f . It is clear that when q = 29+, E Kb becomes smaller than E n and nuclear excitation induced by a K-capture is possible. q indicates the ion charge state before the electron capture. Thenumber of displayed channels for q = 29+ and 30+ has been considerably reduced to improve readability. NEEC IN LASER-PLASMA SCENARIO WITH GSA
Following the theory presented in Ref. 4, it is possible to compare the reaction rates provided by NEEC and theprocess of direct photoexcitation. In particular, considering the parameters reported in Table III and using the firstscaling law of Ref. 20, we obtain T e = 2 . n e = 9 × cm − as plasma temperature and electron density,respectively. TABLE III. Laser characteristics and absorption coefficient. E pulse . R focal . µ m τ pulse
50 fs λ
800 nm f By means of the radiative collisional code FLYCHK [21] it is possible to determine the ion charge state distribution,considering a non-local thermodynamical equilibrium steady state for the plasma. Fig. 6 shows the results obtainedfor the electron flux I e and the charge state distribution P q considering this scenario. q Ge ba FIG. 6. (a) Relativistic electron flux distribution produced by a 2 . q = [23+ , Continuing the derivation of Eq. 6, the partial NEEC rate for the capture level α r in an ion characterized by a chargestate q can be written as [4]: λ q,α r NEEC = S q,α r NEEC I e ( E α r ) , (15)where E α r represents the resonance energy of the capture channel α r and I e the electron flux. Considering NEEC intoions which are in their ground state (GSA), the total NEEC rate can be written as summation over all the capturechannels α r and all over the charge state q present in the plasma: λ NEEC = X q X α r P q λ q,α r NEEC . (16)As evidenced in Fig. 6b, more than 99 .
6% of the ions fall in the range q = [23+ , λ NEEC = 1 . × − s − , while for direct photoexcitationthe total rate is λ γ = 6 . × − s − . The strongest resonance channel of Table II is contributing to the partialNEEC rate as λ , / NEEC = 1 . × − s − and P q = 2 . × − . Even if P q of this channel would have been 1, itwould have been not enough to compensate the photoexcitation rate λ γ .1In case of NEEC-EXI, it is necessary to sum also all over the initial electronic configurations α , thus Eq. 16modifies as following: λ NEEC = X q X α X α r P α q λ q,α ,α r NEEC . (17)A direct comparison with NEEC-EXI rate is not straightforward and beyond the scope of this paper. Indeed, adynamical study of the plasma formation and expansion is needed through particle-in-cell (PIC) codes. Nonetheless, itis worth to mention that for the resonance energy of the K-channels the electron flux provided by the plasma, evaluatedin Fig. 6a, is few orders of magnitude less than that for the L-channels range. In this case, either a more proper choiceof the electron temperature has to be done, or a free electron source in resonance with the desired channels has tobe provided. For example, considering the 1s / electronic configuration at q = 29+, assuming P α q ≃ × cm − s − eV − at low energy (e.g., with an optimized external electron source), the totalNEEC rate could reach the value of λ NEEC = 3 . − for the highest resonance strength of the K-shell, significantlyhigher than the one obtained with the direct photoexcitation. In these circumstances, the external source can act asan electron switch that boosts the isomer depletion. ∗ simone.gargiulo@epfl.ch † fabrizio.carbone@epfl.ch[1] S. S. Wong, Introductory nuclear physics (John Wiley & Sons, 2008).[2] A. Zadernovsky and J. Carroll, Hyperfine interactions , 153 (2002).[3] M. Harston and J. Chemin, Physical Review C , 2462 (1999).[4] J. Gunst, Y. Wu, C. H. Keitel, and A. P´alffy, Physical Review E , 1 (2018), arXiv:arXiv:1804.03694v1.[5] J. Oxenius, Springer Series in Electrophysics, Berlin: Springer, 1986 (Springer Science & Business Media, 1986).[6] T. Kibedi, T. Burrows, M. B. Trzhaskovskaya, P. M. Davidson, and C. W. Nestor Jr, Nuclear Instruments and Methodsin Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment , 202 (2008).[7] J. Rzadkiewicz, M. Polasik, K. S labkowska, L. Syrocki, E. W¸eder, J. J. Carroll, and C. J. Chiara,Phys. Rev. C , 044309 (2019).[8] M. Rysavy and O. Dragoun, arXiv preprint nucl-ex/0006009 (2000).[9] G. Gosselin and P. Morel, Physical Review C , 064603 (2004).[10] F. Larkins, Atomic Data and Nuclear Data Tables , 311 (1977).[11] M. F. Gu, Canadian Journal of Physics , 675 (2008).[12] P. V. Bilous, G. A. Kazakov, I. D. Moore, T. Schumm, and A. P´alffy, Physical Review A , 1 (2017).[13] A. P. Yutsis, V. Vanagas, and I. B. Levinson, Mathematical apparatus of the theory of angular momentum (Israel programfor scientific translations, 1962).[14] L. C. Biedenharn and J. D. Louck,
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