Calculations of nuclear excitation by electron capture (NEET) in nonlocal thermodynamic equilibrium plasmas

PHYSICAL REVIEW C 81, 034609 (2010) Calculations of nuclear excitation by electron capture (NEET) in nonlocal thermodynamic equilibrium plasmas P. Morel, V. Méot, G. Gosselin, G. Faussurier, and C. Blancard CEA, DAM, DIF, F-91297 Arpajon, France (Received 14 September 2009; published 23 March 2010) The nuclear excitation by electron capture (NEET) process may occur when the energy differences between two nuclear levels and between two electronic states are nearly equal, provided the quantum selection rules are fulfilled. These resonant conditions drastically limit the number of possible candidates, even though thermodynamic conditions encountered in hot dense plasmas do modify the orbital electronic binding energy and the resonance conditions. 201 Hg, with a low-lying isomeric state located at 1.565 kev, can be excited by NEET process in a laser-created plasma. However, its correct calculation requires nonlocal thermodynamic equilibrium (non-) atomic physics treatment because current laser-created plasmas do not reach high-enough temperature in the area at. In this article, we describe the calculation leading to an estimated excitation rate and discuss the influence of /non- physics with an average-atom model and the use of a Gaussian variance calculation to estimate the broadening around the mean energy mismatch. DOI: 10.1103/PhysRevC.81.034609 PACS number(s): 23.20.Nx, 24.10. i I. INTRODUCTION Hot plasmas created in the stellar environment or by lasermatter interaction contain nuclei in excited states generated by electromagnetic processes. The excitation rate calculations must take into account each process [1]. In plasma, the most important electromagnetic processes are photon absorption, inelastic electron scattering [2], nuclear excitation by electron capture (NEEC) [3], and, finally, nuclear excitation by electron transition (NEET) [4]. The occurrence of these processes can drastically modify the nuclear lifetime by several orders of magnitude compared to an isolated atom [5]. Experimental investigation of nuclear excitation by electronic processes in plasma requires high-power lasers. Such facilities create plasmas at temperatures around several kev. Thus, heavy ions with low-energy nuclear levels can be significantly excited. Among those processes, NEET is the most efficient nuclear excitation mechanism. In this article, we focus on this specific phenomenon because it is the most promising process to put in evidence nuclear excitation in a laser-produced plasma. Describing the NEET process in plasma requires a detailed knowledge of the involved atomic physics. For an excited high-z ion and a specified charge state, there exists a huge amount of electronic configurations whose complete description is beyond the current computational capabilities. The average-atom model [6] is a convenient way to cope with such number of configurations by featuring mean noninteger electronic subshell populations. It is possible to evaluate the probability of having both at least an electron on an outer shell and a hole on an inner shell such so that the electronic transition energy matches with the nuclear transition energy. Provided fulfillment of quantum rules, a NEET rate can be derived. A model using the average atom approach at local thermodynamic equilibrium () has been developed and applied to the excitation of the first isomer level of 235 U[4] (76.8 ev) and 201 Hg (1.565 kev) in plasma [7]. A NEET process with 201 Hg may occur for several transitions involving electrons from shells n = 6ton = 4(n represents the principal quantum number). We showed in [7] that ionizations Q = 42 + and 44 + are required to get the lowest temperature transitions, respectively N1-P 1 (4s 1/2 6s 1/2 ) and N2-P 3 (4p 1/2 6p 3/2 ), resonant. Unfortunately, in a current laser-produced plasma, such charge states are encountered in a region of the thermodynamic diagram where atomic physics is not at [8]. To describe matter in the non- regime, we use an average-atom code based on a screened hydrogenic model (SHAAM) [9]. The nonrelativistic average-atom model SHAAM offers the opportunity to calculate self-consistently under local and/or nonlocal thermodynamic equilibrium conditions. However, as NEET coupling matrix elements depend strongly on the radial electronic wave functions close to the nucleus, one must take into account the relativistic behavior of the electrons in our calculation. Therefore, calculating the electron wave functions requires solving the Dirac equation. Because developing a fully non- relativistic average-atom model is a difficult task, we propose a scheme to use the nonrelativistic SHAAM electronic shell occupations to obtain the required electron wave functions from a Dirac equation solver. One main goal of this article is to provide realistic estimation of nuclear excitation rates in plasma to design an experiment adapted to a laser facility. We first describe the average-atom model SHAAM. Then, we present the NEET rate calculation for 201 Hg and explain how to determine each physical quantity of interest. To check the consistency of our approach, we performed some calculations under physics to compare with our [4]. Then we show various results of mismatch, ionization state, and NEET rate under both and non- conditions. II. THE SHAAM MODEL The description of non- plasmas is very difficult owing to the huge number of electronic configurations that must be 0556-2813/2010/81(3)/034609(7) 034609-1 2010 The American Physical Society

P. MOREL et al. PHYSICAL REVIEW C 81, 034609 (2010) taken into account. To bypass this problem, a very powerful solution is to consider only one configuration, that is, the electronic configuration of the average-atom model [9]. This model is attractive and has been proven to be well-defined and consistent with [9]. The SHAAM is an extension to non- of the average-atom model [10]. In short, this approach consists of finding an approximate solution to the rate equation written in the space of electronic configurations. This solution is found assuming the density probability of the electronic configurations to be a Gaussian distribution centered around a reference configuration, that is, the average-atom configuration. The noninteger subshell occupation numbers of this configuration and the inverse matrix (C kk ) of the real symmetric matrix, which defines the Gaussian density probability, satisfy a set of nonlinear coupled time-dependent equations that must be solved self-consistently. These equations are furthermore under constraint because the neutrality of the plasma must be ensured. This inverse matrix (C kk ) has also an immediate interpretation because it gives an estimate of the electron correlation matrix [11]. We can thus use the classical theory of fluctuations extended to non- conditions to calculate the variance of any physical quantity, which is an explicit function of the average-atom populations [10,12,13]. Atomic structure is described using a nonrelativistic screened-hydrogenic model with l splitting [14]. Transition rates are given by analytical expressions [9]. This SHAAM model is very fast to solve and can be used to describe various non- conditions. In this work, we restrict ourselves to steady-state conditions. Dense plasma effects are very important in both and non- regimes. In this article, we have chosen a degeneracy-lowering formula able to keep, for each density-temperature couple, the same charge state as those of the [4]. The general expression of the degeneracy D nl, for principal quantum number n and orbital number l, is written as follows [15,16]: 4l + 2 D nl = 1 + [ a a ] 0(nl) b, (1) R 0 where a 0 (nl) and R 0 denote the Bohr orbit radius of level nl and the ionic sphere radius, respectively. The chargestate consistency between former and present calculations is reached with a = b = 3. III. NON- NEET CALCULATION A. NEET rate For a given plasma temperature and density, we derived the NEET rate expression [4]: (ρ,t e ) = 2π h D 1p 1 (1 p 2 ) R 1,2 2 1 e δ2 2σ 2. (2) 2πσ 2 In this expression, p 1 and p 2 denote the occupation probability of the initial and final atomic subshell (indices 1 and 2, respectively); D 1 is the degeneracy of the upper electron subshell; thus, D 1 p 1 represents the corresponding number of electrons; σ is the standard deviation of the Gaussian envelope representing the spread of configuration energies beyond the average atom, centered on the mean configuration; δ is the corresponding energy mismatch, defined as the energy difference between two atomic configurations and two nuclear levels; and R 1,2 is the atom-nucleus coupling matrix element. Even though this NEET rate expression was derived within an context, no specific hypothesis is required. Therefore, we extend its use in this article in a non- framework where we have to re-evaluate each of the populations, the mismatch, and its variance. B. Mismatch variance The NEET rate depends strongly on the mismatch between the nuclear and the electronic transition energies. However, the average-atom model is intrinsically unable to predict the distribution of the mismatch over the huge number of electronic configurations. Estimating this distribution variance can be done with the classical theory of fluctuations. This gives a general expression of the variance dependent on the subshell population, σ 2 δ = (δ δ)2 = k,k δ δ C kk, (3) N k N k where C kk = N k N k and N k = N k N0 k is the population difference of the subshell k between the real and the average configurations. The mismatch δ is written δ αβ = E(C α ) E(C β ) E nuc, (4) where E nuc is the nuclear transition energy and the energy of a real configuration is related to the average configuration energy, E(C), by ( ) E(C) = E(C) + ε k Nk Nk 0 + 1 2 kk k=1 ( ) ( ) Nk Nk 0 Vkk Nk Nk 0, (5) where V kk is the electrostatic matrix that contains a Coulombian component of direct interaction and exchange between a k-subshell electron and a k -subshell electron. ε k denotes the monoelectronic energy of a k-subshell electron. The mismatch δ is then written δ αβ = δ 12 + (V 1k V 2k ) ( ) N k Nk 0, (6) k=1 where δ 12 is the average-atom mismatch. Inserting expression (6) into Eq. (3), the final variance equation becomes σ 2 = kk (V 1k V 2k )(V 1k V 2k ) C kk. (7) 034609-2

CALCULATIONS OF NUCLEAR EXCITATION BY... PHYSICAL REVIEW C 81, 034609 (2010) C. relativistic calculation B. Rozsnyai [6] developed a relativistic average-atom model at based on the density functional theory for describing the exchange and correlation effects. It is based on an iterative procedure where the initial bound and free electron densities and a chemical potential are first assumed. An atomic potential, including nuclear, electronic, exchange and correlation components, as a function of the radial variable, can be generated. Then, the Dirac equation is solved for all bound electrons. Electroneutrality is assumed inside the Wigner-Seitz cell. The wave functions and energies of all the occupied bound states are calculated. The subshell populations are derived using the Fermi-Dirac statistics. This is the point where the hypothesis applies. This ends the first iterative loop by providing an improved bound and free electron distributions and a chemical potential. The process is iterated until the atomic potential converges. In our calculation, we use the non- populations N nl obtained in the nonrelativistic SHAAM model. They are split between both spin states proportionally to the subshell degeneracy: 2j + 1 N nlj = N nl 4l + 2. (8) The same principle stands for the correlation matrix elements: C nlj,n l j = 2j + 1 2j + 1 4l + 2 4l + 2 C nl,n l. (9) These populations are then used to calculate a single iteration identical to a relativistic average atom model iteration. This provides the relativistic wave functions of all orbitals, the chemical potential, and the average electronic potential, which are self-consistently calculated for an electronic temperature T e and density ρ. It is pointed out that the electronic wave functions are solutions of the Dirac equation. That way, every parameter of Eq. (2) is available to estimate the non- NEET rate in a large density and temperature domain. IV. RESULTS As the SHAAM model is consistent with, we compare in this section its results with our previous calculations at [4,7]. Three main differences exist between the two models: different electronic structure treatment, relativistic versus nonrelativistic populations, and density-effect treatments. We present in Fig. 1 the average occupation of both 4s and 6s atomic shells of 201 Hg, as a function of electronic temperature. These shells are those involved in the lowest-temperature NEET transition. Both shells occupation results exhibit a rather good agreement. However, the occupations in the are always slightly lower than in those the previous work: Binding energies of relativistic electrons are greater than those of nonrelativistic electrons, leading to more populated electronic shells. Density effects are usually more important for outer shells. The small discrepancy, less than a factor of 3, occurring at low temperature for the 6s subshell, can be explained by the treatment of density effects, which is different between the previous and the present calculations. Sub-shell occupation 10 0 10-1 10-2 10-3 10-4 6s 1/2 4s 1/2 10-5 FIG. 1. 201 Hg occupations of the N1 (4s 1/2 )andp 1(6s 1/2 )at 10 2 g/cm 3, as a function of temperature, from previous and present work, under treatment. Figure 2 shows the mismatch as a function of temperature near resonance between previous and, for the 4s 6s resonant transition. Good agreement is found between both calculations. In a more realistic plasma description, an atomic transition is split into a huge number of individual electronic transitions whose energies depend on the detailed configurations leading to statistical broadening of the mismatch. In our model, this distribution is represented by a Gaussian function whose standard deviation σ [Eq. (7)] of the mismatch [Eq. (6)] is shown in Fig. 3. The general shapes are very similar but the positions of the extrema are slightly shifted toward low temperatures. The higher the temperature, the larger the shift. Figure 4 shows the NEET rates, as a function of temperature. The local maxima, each related to a set of resonant transitions, are located nearly at the same temperatures. The little resonant shifts are more discernible at higher temperatures and are explained by small shifts of the mismatch. All these results from our previous model and the current SHAAM approach are very similar. Thanks to the δ (ev) 0-500 -1000-1500 δ (ev) 200 100-100 -200 0.1 0.2 0.3-2000 FIG. 2. 201 Hg mismatch δ at 10 2 g/cm 3, as a function of temperature, for the 4s 1/2 6s 1/2 transition, from previous and present works. 0 034609-3

P. MOREL et al. PHYSICAL REVIEW C 81, 034609 (2010) 80 70 σ (ev) Ionization 60 50 40 30 FIG. 3. Standard deviation of the mismatch, as a function of temperature, for the 4s 6s NEET transition of 201 Hg at 10 2 g/cm 3, from previous and, under conditions. non- consistency of the SHAAM model, we can safely ascribe the results presented later in this article solely to non- effects. V. RESULTS In this section, we show the influence of the non- treatment on the NEET rate. Under the non- conditions, dealt within the SHAAM model, we have different radiative and electronic temperatures. The radiative temperature is the temperature of the radiation field supposed to be described by a Planck distribution function at this temperature. Hereafter, we suppose a constant radiative temperature of 100 ev. It roughly corresponds to what can be encountered in plasma conditions created by a direct-drive 4 W/cm 2 nanosecond laser impact on a high-z target in the conversion zone. In our calculations, so long as radiative temperature is below electronic temperature, transition rates are not notably 20 Electronic temperature (kev) FIG. 5. and non- average ionization degree, as a function of temperature, for 201 Hg at 10 2 g/cm 3. modified, and in any case, the radiative temperature never exceeds electronic temperature during laser-matter interaction. We present, in Fig. 5, 201 Hg and non- ionization at 10 2 g/cm 3, as a function of electronic temperature. The non- average ionization degree is lower than the one at given electronic temperature and density owing to the fact that recombination processes are not fully counterbalanced by ionization processes in non- situations. Moreover, ionization increases much more slowly with electronic temperature under non- conditions than in the case. We present, in Fig. 6, the and non- occupations of the 4s and 6s subshells at 10 2 g/cm 3 as a function of electronic temperature. The occupation decreases much more slowly with the electronic temperature under non- treatment. For each subshell, non- occupation numbers are higher than ones at given electronic temperature and density. This is an expected consequence of the lower ionization at non-. We present, on Fig. 7, and non- NEET rates, as functions of electronic temperature, obtained 10 5 λ exc Sub-shell occupation 10 0 10-1 10-2 10-3 10-4 4s 1/2 6s 1/2 FIG. 4. NEET rate under conditions, as a function of temperature, for 201 Hg at 10 2 g/cm 3, from previous and present work. 10-5 FIG. 6. 201 Hg occupations of the 4s 1/2 and 6s 1/2 subshells, at 10 2 g/cm 3, as a function of electronic temperature, with and non- treatments. 034609-4

CALCULATIONS OF NUCLEAR EXCITATION BY... PHYSICAL REVIEW C 81, 034609 (2010) 10 5 0.30 σ (kev) 0.25 0.20 0.15 0.10 0.05 N1_P1 N2_P3 N1_P1 N2_P3 ρ = 10-2 g/cm 3 Electronic temperature (kev) FIG. 7. 201 Hg NEET rate comparison, and non-, at 10 2 g/cm 3, as a function of electronic temperature. for 201 Hg at 10 2 g/cm 3. The non- rate exhibits a slower increase and reaches a first maximum at a higher temperature, around 1 kev. This first maximum corresponds to both the 6s 4s and 6p 3/2 4p 1/2 resonances. This higher resonance temperature is a direct consequence of the lower ionization and higher occupations shown in Figs. 5 and 6, respectively. This seems to indicate that the average charge state might be a better parameter than the corresponding electronic temperature for comparing both results. Therefore, we show, in Fig. 8, the and non- rates as functions of average charge state. This time, the first maximum, dominated by both the 6s 1/2 4s 1/2 and the 6p 3/2 4p 1/2 transitions, is located at nearly the same charge state (42 + to 44 + ), even if a higher temperature is needed to reach this ionization. However, the width of the first maximum is wider in the non- condition than in the one. This originates in the correlation matrix elements of the transition involved shells, and, subsequently, in the corresponding standard deviation. We present, in Fig. 9, the standard deviation, for both N1-P 1(4s 1/2 6s 1/2 ) and N2-P 3 (4p 1/2 6p 3/2 ) transitions, as a function of the average charge state. We clearly see a sharp decline of σ close to n = 3 10 5 ρ = 10-2 g/cm 3 30 40 50 60 70 80 Average charge state FIG. 8. and non- 201 Hg NEET rates, at 10 2 g/cm 3,asa function of average charge state. 0.00 20 30 40 50 60 70 80 Average charge state FIG. 9. and non- 201 Hg standard deviation, for the N1- P 1 atomic shells at 10 2 g/cm 3, as a function of average charge state. or M shell ( Q =70) and n = 4orN shell ( Q =52) closures (tightly bound electrons) and a maximum roughly half way between two consecutive shell closures (loosely bound electrons). The lack of interaction in the non- condition (compared to regime) corresponds to a more peaked shape of the standard deviation. The first maximum of the NEET rate, shown on Fig. 8, is located around Q =44, where the standard deviation difference between and non- is rather important. Thus, the non- excitation rate maximum is wider. The second maximum of the NEET rate is located in a average charge-state area ( Q around 50) where both and non- standard deviations are nearly the same and we do not observe a notable width difference. The non- approach is a time-consuming calculation, in particular close to subshell closure, where convergence difficulties may occur. So, it is possible to evaluate an ionization temperature T z, defined as the electronic temperature that gives the same charge state in as the non- charge state at given density. This approach is consistent with by construction, although this notion of effective temperature lacks a clear theoretical justification. This kind of approximation is commonly used in the Busquet model [17] and has the paramount advantage of requiring much shorter calculation time. In Fig. 10, we plot both interpolated non- NEET rates obtained with an ionization temperature and non- calculations, as a function of non- electronic temperature, for two densities of 10 3 and 10 2 g/cm 3.It clearly shows a poor agreement between non- T z and fully non- calculations, especially for low densities far from the conditions. The notion of ionization temperature seems seductive but should be used with caution in this practical application. All these results have been obtained with a fixed radiative temperature T r = 100 ev which lies within the range of radiative temperatures that can be encountered in laser-induced plasmas able to reach the thermodynamic conditions presented below. However, it is important to check the influence of T r on the NEET rate. This has been done on Fig. 11 for three radiative 034609-5

P. MOREL et al. PHYSICAL REVIEW C 81, 034609 (2010) ρ=10-2 g/cm 3 10-2 g/cm 3 ρ=10-3 g/cm 3 TZ Tr=200eV Tr=100eV Tr=10eV 0 1 2 3 4 5 6 Electronic temperature (kev) FIG. 10. and with extrapolated ionization temperature NEET rates for 201 Hg at 10 3 and 10 2 g/cm 3 as a function of electronic temperature. 35 40 45 50 55 Average charge state FIG. 11. (Color online) 201 Hg NEET rates, as functions of the average charge state, for radiative temperatures of 10, 100, and 200 ev. temperatures: 10, 100, and 200 ev, as a function of the average charge state. No difference can be established below 100 ev and some discrepancies appear at 200 ev, especially for low charge states. These NEET rate estimations could be used to design a laser-driven experiment. A high-power laser beam might be able to excite a significant number of 201 Hg. For instance, an intensity of 4 W/cm 2 could be reached under the following conditions: nanosecond pulse duration, focal spot diameter around 350 µm, and 100 J energy. Under these conditions, a hydrodynamic-radiative simulation predicts, at the end of the laser pulse, several 4 201 Hg nuclei with a charge state beyond 42 + and a charge-state distribution up to 50 + in the area where collisional-radiative equilibrium takes place (density range, 10 3 toafew10 2 g/cm 3 ). Such conditions offer the possibility for 201 Hg nuclei to be excited in a wide range of NEET resonant transitions. If one considers that mercury ions remain in such charge states as long as 10 ps, we can finally obtain several million 201m Hg ions by NEET process per laser shot on a highly enriched target. The detection of these isomers is closely connected to their own decay, essentially by internal conversion. For a charge state beyond 30 +, the internal conversion is inhibited and the 201m Hg isomer lifetime, of 81 ns for neutral atom [7], is highly enhanced up to a millisecond after depopulation of outer electronic shells. The mercury ions are transported from the target to the detector as a beam with a speed between 10 7 and 10 8 cm/s. A millisecond lifetime makes it possible to put the detector far enough from the target and its heavily noisy environment. VI. CONCLUSION In this article, we extended the formalism of a previous work [4] in a non- environment by inserting non- occupation numbers obtained from a self-consistent non- average-atom approach based on a screened-hydrogenic model. The other ingredients, such as chemical potential and wave functions, were then reprocessed in our relativistic average-atom model based on the density functional theory, to solve the Dirac equation and then derive a non- NEET rate. Special focus has been put on the correlation matrix to deal with the huge number of electronic configurations around the average mismatch. A more complete calculation would consist of a multiconfiguration approach, taking into account intermediate coupling, for each repartition of electrons on the subshells. However, taking into account the huge number of such electronic configurations, even if restricted to a single charge state, remains beyond the current computing capacities. After checking consistency between our previous results and this new model at, we performed extended calculations of the non- NEET rate of the first excited state of 201 Hg as a function of electronic temperature for a given radiative temperature of 100 ev. The lower photon interaction rate in the plasma leads to a lower ionization, which in turn shifts the NEET resonances to higher electronic temperatures. If presented as a function of average charge state, both and non- rates exhibit similar behavior. This result confirms the strong dependence of the NEET rate on atomic populations. Thanks to statistical mechanics, we showed that it is possible to calculate plasma non- NEET rates. [1] M. R. Harston and J. F. Chemin, Phys. Rev. C 59, 2462 (1999). [2] G. Gosselin, N. Pillet, V. Méot, P. Morel, and A. Dzyublik, Phys. Rev. C 79, 014604 (2009). [3] G. Gosselin and P. Morel, Phys. Rev. C 70, 064603 (2004). [4] P. Morel, V. Méot, G. Gosselin, D. Gogny, and W. Younes, Phys. Rev. A 69, 063414 (2004). [5]G.Gosselin,V.Méot, and P. Morel, Phys. Rev. C 76, 044611 (2007). [6] B. Rozsnyai, Phys. Rev. A 5, 1137 (1972). 034609-6

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