Nuclear excitation by electron capture in optical-laser-generated plasmas
Jonas Gunst, Yuanbin Wu, Christoph H. Keitel, Adriana Pálffy
NNuclear excitation by electron capture in optical-laser-generated plasmas
Jonas Gunst, ∗ Yuanbin Wu, † Christoph H. Keitel, and Adriana P´alffy ‡ Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany (Dated: June 20, 2018)The process of nuclear excitation by electron capture in plasma environments generated by theinteraction of ultra-strong optical lasers with solid-state samples is investigated theoretically. Withthe help of a plasma model we perform a comprehensive study of the optimal parameters for mostefficient nuclear excitation and determine the corresponding laser setup requirements. We discernbetween the low-density plasma regime, modeled by scaling laws, and the high-density regime, forwhich we perform particle-in-cell calculations. As nuclear transition case study we consider the 4.85keV nuclear excitation starting from the long-lived
Mo isomer. Our results show that the optimalplasma and laser parameters are sensitive to the chosen observable and that measurable rates ofnuclear excitation and isomer depletion of
Mo should be already achievable at laser facilitiesexisting today.
I. INTRODUCTION
The invention of the laser more than 50 years ago [1]revolutionized atomic physics, leading to the better un-derstanding and control of atomic and molecular dynam-ics. Covering several frequency scales, intense coherentlight sources available today open unprecedented possi-bilities for the field of laser-matter interactions [2] alsobeyond atomic physics. Novel x-ray sources as the X-rayFree Electron Laser (XFEL) open for instance new pos-sibilities to drive low-lying electromagnetic transitionsin nuclei [3]. On the other hand, high-power opticallasers with their tremendous efficiency in transferring ki-netic energy to charged particles may cause formationof plasma [4], the host of complex interactions betweenphotons, electrons, ions and the atomic nucleus. Nuclearexcitation in optical laser-generated plasmas has been thegeneral subject of several studies so far [5–18], while thepossibilities to induce nuclear transitions to low-lying ex-cited levels using high-energy lasers have been summa-rized in Refs. [19–23]. In addition, nuclear excitationmechanisms in cold high-density plasmas generated bythe interaction of XFEL sources with solid-state targetswere investigated in Refs. [24, 25].Nuclear excitation may occur in plasmas via severalmechanisms. Apart from direct or secondary photoex-citation, the coupling to the atomic shell via processessuch as nuclear excitation by electron transition (NEET)or electron capture (NEEC) [26, 27] may play an impor-tant role. In particular, it was shown that as secondaryprocess in the plasma environment, NEEC may exceedthe direct nuclear photoexcitation at the XFEL by ap-proximately six orders of magnitude [24, 25]. Since thenuclear coupling to the atomic shell is generally speak-ing very sensitive to the plasma conditions, the ques-tion raises whether one can tailor the latter for maxi-mizing the effect of NEEC. While the tunability of the ∗ [email protected] † [email protected] ‡ palff[email protected] XFEL-generated plasma properties are limited due to thespecific properties of x-ray-atoms interaction, high-poweroptical lasers are able to generate plasmas over a broadparameter region as far as both temperature and plasmadensity are concerned. In a very recent letter, NEEC wasshown to be the dominant nuclear excitation mechanismfor a broad parameter range in optical-laser-generatedplasmas, and results on tailored optical laser parametersfor maximizing NEEC were presented [28].In this work, we present a systematic theoretical studyof NEEC in optical-laser-generated plasmas. Having theconcrete scenario of ultra-short optical laser-generatedplasmas in mind, we develop a plasma model that canbe easily applied to any nuclear parameters. Based onthis model we deduce the optimal NEEC parameters interms of plasma temperature and density. Due to thecomplexity of the processes involved, we show that theplasma parameters for maximal NEEC rate are not iden-tical to the ones that determine the maximal numberof excited nuclei. We then further investigate how theoptimal NEEC parameters in the temperature-densitylandscape can be accessed by a short and intense laserpulse considering various experimental conditions suchas laser intensity, wavelength, pulse duration or pulseenergy. We discern in our treatment between two cases,the low-density plasma regime, modeled by the scalinglaw, and the high-density regime, where we use dedicatedparticle-in-cell (PIC) simulations.As case study we consider the 4.85 keV nuclear tran-sition starting from a long-lived excited state of Moat 2.4 MeV. Such states are also known as nuclear iso-mers [29] and have been the subject of increased attentiondue to the potential storage of large amounts of energyover long periods of time [30–36]. For
Mo, the addi-tional 4.85 keV nuclear excitation leads to the depletionof the isomer and release on demand of the stored energy.Apart from this appealing scenario,
Mo is interestingalso because of the recently reported observation of iso-mer depletion via NEEC of the 4.85 keV transition [37].Our results show that for high electron densities, NEECis actually the dominant nuclear excitation channel for
Mo, surpassing by orders of magnitude photoexcita- a r X i v : . [ phy s i c s . p l a s m - ph ] J un tion in the plasma. Surprisingly, a six orders of mag-nitude increase in the number of excited nuclei can beachieved employing a high-power optical laser comparedto the previously investigated case of XFEL-generatedplasmas. The calculated maximal number of depletedisomers for realistic laser setup parameters appears tohave reached values large enough to be observable in anexperiment. Although still far from the final goal, this isa further milestone on the way to the realization of con-trolled energy storage and release via nuclear isomers.This paper is structured as follows. After introducingthe theory for NEEC and nuclear photoexcitation in aplasma environment and the employed plasma model inSection II, we start by investigating the optimal plasmaconditions for NEEC in terms of electron density andtemperature based on a simplified model for sphericalplasmas in Section III. As part of this Section, we dis-cuss the influence of ionization potential depression, theexpected contribution from photoexcitation and a hydro-dynamic model for plasma expansion. Section IV is thendevoted to optical laser-generated plasmas. Since theplasma generation involves different processes depend-ing on the electron density, we divide this Section intotwo parts, IV A for low density and IV B for high den-sity plasmas, respectively. As a main result we evaluatethe optimal laser parameters for low-density scenarios atthe example of high-power laser facilities. Surprisingly,our analysis of a PIC simulation for high electron den-sities shows that similar nuclear excitation numbers canbe achieved with 100 J lasers available at many facili-ties around the world. The paper ends with concludingremarks. II. THEORETICAL APPROACH
This Section introduces the theoretical approach usedfor describing nuclear excitation in the plasma. Afterfirst general considerations on the setup, we will sketchour calculations for NEEC and photoexcitation rates inthe plasma and outline the used plasma model. Atomicunits (cid:126) = m e = e = 4 πε = 1 are used throughoutsubsections II B and II C. A. General considerations
We investigate the interaction of ultra-strong laserswith a solid-state target. The strong electromagneticfield of optical lasers leads to field ionization and accel-eration of the electrons in the target. The acceleratedhot electrons can lead to further ionization of the targetvia collisional ionization. Since the laser wavelength isin the optical range, the electronic heating occurs essen-tially at the surface of the material. The hot electronsproduced by the laser are accelerated away from the in-teraction region generating an electric field due to thecharge separation. Attracted by this electric field the ions subsequently follow the hot electrons, resulting inthe formation of a neutral plasma surrounding the tar-get. The time scale of the plasma generation process ison the order of the pulse duration (plus the time for theacceleration of the ions). Moreover, near the focal spotof the laser the plasma can be considered as uniform interms of electron density and temperature.Looking in more detail at the start of the laser-targetinteraction, there will be a pre-pulse leading to the gen-eration of a pre-plasma. The main pulse thus interactswith this cold pre-plasma instead of the initial target.Consequently, the final plasma consists of a cold (com-ing from the pre-pulse) and a hot electron distribution(coming from the main pulse). The cold electron distri-bution stays essentially in the region around the inter-action point with the target. As far as we consider thintargets ( ∼ µ m) this volume is small in comparison to thetotal plasma volume, such that the cold electrons can beneglected.In order to reach high electron densities ( ∼ cm − )thicker targets need to be considered. Analogously to thecase of thin targets, the electromagnetic laser field accel-erates the electrons of the target surface into the innerregion of the target. These hot electrons then subse-quently lead to further collisional ionization events insidethe target which can result in a very dense plasma in thisregion [38, 39]. However, the absorption fraction f seemsto be effectively much lower for this heating mechanismin comparison to the thin target case.For our particular study of nuclear excitation in plas-mas, we consider a strong optical laser that interacts witha solid-state target containing a fraction of nuclei in theisomeric state. NEEC and/or photoexcitation may oc-cur in the generated plasma. In the resonant process ofNEEC, a free electron recombines into a vacant boundatomic state with the simultaneous excitation of the nu-cleus. The isomers can then be excited to a trigger statewhich rapidely decays to the nuclear ground state andreleases the energy “stored” in the isomer. For Mothe stored energy corresponds to the isomeric state at2424 keV. A further 4.85 keV excitation leads to the fastrelease within approx. 4 ns via a decay cascade contain-ing a 1478 keV photon which could serve as signature ofthe isomer depletion. Recent experimental results on theisomer depletion of
Mo in a channeling setup have de-duced a rather high NEEC rate of the 4.85 keV transition[37] encouraging further studies on this nuclear transi-tion.
B. NEEC in plasma environments
In the plasma, free electrons with different kinetic en-ergies are available. At the NEEC resonant energy, elec-trons may recombine into ions leading to nuclear excita-tion. The resonance bandwidth is determined by a nar-row Lorentz profile. Since the kinetic energy of free elec-trons in a plasma is distributed over a wide range, manyresonant NEEC channels may exist. In the following wewill shortly describe how such a situation can be handledtheoretically in terms of reaction rates.In order to restrict the number of possible initial elec-tron configurations, for a lower-limit estimate, we con-sider in the following only NEEC into ions which are intheir electronic ground states. In this case, the initialelectronic configuration α is uniquely identified by thecharge state number q before electron capture. In theisolated resonance approximation, the total NEEC reac-tion rate in the plasma can be written as a summationover all charge states q and all capture channels α d , λ neec ( T e , n e ) = (cid:88) q (cid:88) α d P q ( T e , n e ) λ q,α d neec ( T e , n e ) . (1)Here, P q is the probability to find in the plasma ionsin the charge state q as a function of electron tempera-ture T e and density n e . The partial NEEC rate into thecapture level α d of an ion in the charge state q can beexpressed by the convolution over the electron energy E of the single-resonance NEEC cross section σ i → dneec and thefree-electron flux φ e , λ q,α d neec ( T e , n e ) = (cid:90) dE σ i → dneec ( E ) φ e ( E, T e , n e ) . (2)The NEEC cross section σ i → dneec as a function of the freeelectron energy E is proportional to a Lorentz profile L d ( E − E d ) centered on the resonance energy E d . Thewidth of the resonance is typically determined by thenuclear linewidth of approx. 100 neV. Since over thisenergy scale φ e can safely be considered as constant, wecan approximate the Lorentz profile by a Dirac-delta-likefunction. The match between the resonance energy E d and the functional temperature dependence of the elec-tron flux φ e ( T e ) determines the quantitative contributionof the individual NEEC channels.The electron flux φ e in the plasma can be written asthe product of the density of states g e ( E ), the Fermi-Dirac distribution f FD ( E, T e , n e ) for a certain electrontemperature T e and the velocity v ( E ), φ e ( E, T e , n e ) dE = g e ( E ) f FD ( E, T e , n e ) v ( E ) dE . (3)The temperature dependence of φ e is only included inthe Fermi-Dirac statistics f FD [40]. The density of statesas well as the velocity are determined by considering therelativistic dispersion relation for the free electrons. Theelectronic chemical potential µ e occurring in the expres-sion of f FD is fixed by adopting the normalization (cid:90) dE g e ( E ) f FD ( E, T e , n e ) = n e . (4)Thus, the electron flux in the plasma depends on boththe electron temperature T e and the density n e .The theoretical formalism for the calculation of theNEEC cross section σ neec has been presented elsewhere [24, 25, 41, 42]. The cross section is connected to themicroscopic NEEC reaction rate Y neec via σ i → dneec ( E ) = 2 π p Y i → dneec L d ( E − E d ) , (5)with p the free electron momentum. Substituting Eq. (5)into Eq. (2), the integral over the kinetic electron energy E can be solved by assuming that the free electron mo-mentum p and the NEEC rate Y neec are constant over thewidth of the Lorentz profile L d ( E − E d ), which is the casefor a wide spectrum of isotopes. Eq. (2) then simplifiesto λ q,α d neec ( T e , n e ) = 2 π p Y i → dneec φ e ( E d , T e , n e ) . (6)The total NEEC rate λ neec in Eq. (1) is thereforestrongly dependent on the available charge states and freeelectron energies which both are dictated by the plasmaconditions. Taking the spatial and temporal plasma evo-lution into account, the total NEEC excitation number N exc is connected to the rate λ neec via N exc = (cid:90) V p d r (cid:90) dt n iso ( r , t ) λ neec ( T e , n e ; r , t ) , (7)where n iso denotes the number density of isomers presentin the plasma. For the further quantitative estimate ofthe occurred nuclear excitation, relevant factors are theinteraction time and the volume over which the interac-tion takes place, considered to be the plasma volume V p .These aspects are detailed in subsection II D. C. Resonant nuclear photoexcitation in plasma
Instead of undergoing NEEC, the nucleus can also beexcited by the absorption of a photon which has to be onresonance with the nuclear transition energy E n . Analogto Eqs. (1) and (2), the excitation rate via photons in theplasma can be expressed as λ γ ( T e , n e ) = (cid:90) σ i → d γ ( E ) φ γ ( E, T e , n e ) dE , (8)with the nuclear photoexcitation cross section σ i → d γ ( E ) = π k A i → d γ L d ( E − E n ), where A i → d γ represents the corre-sponding rate. For the calculation of the photoexcita-tion rate, we have adopted the formalism from Ref. [43]that connects A γ with so-called reduced nuclear transi-tion probabilities. For the latter we employ experimentaldata and/or nuclear model calculations later on.In general, the photon flux and hence the photoexcita-tion rate in the plasma depend on the prevailing plasmaconditions represented by electron temperature and den-sity in Eq. (8). In order to evaluate this dependencefurther, we employ two models for the photon flux φ γ inthe following.First, we assume the photons to be in thermodynamicequilibrium (TDE) with the electrons, such that a black-body distribution is applicable, resulting in the density-independent photon flux φ TDE γ ( E, T e ) dE = c g γ ( E ) f BE ( E, T e ) dE. (9)Here, c is the speed of light, g γ represents the photonicdensity of states and f BE denotes the Bose-Einstein dis-tribution [40], respectively. Substituting Eq. (9) intoEq. (8) leads to the photoexcitation rate under TDE con-ditions λ TDE γ ( T e ) = 2 π k A i → d γ φ TDE γ ( E n , T e ) . (10)In the derivation of Eq. (10), the Lorentzian profile hasbeen approximated by a Dirac-delta-like resonance sincethe nuclear transition width is for the considered plasmatemperatures much smaller than the energy region overwhich the photon flux varies significantly.As a second model, we considered the process ofbremsstrahlung as a potential photon source in theplasma. According to Ref. [5], the photon flux emittedvia bremsstrahlung evaluates to φ B γ ( E, E e , T e , n e ) dE dE e = t i (cid:18) dσ B ( E e ) dE (cid:19) φ e ( E e , T e , n e ) dE dE e , (11)where dσ B ( E e ) /dE denotes the bremsstrahlung cross sec-tion differential in the emitted photon energy E and t i represents the target thickness given in atoms per area.For the calculations later on we consider t i = n i R p , where n i represents the ion number density in the plasma and R p the plasma radius. Employing Eq. (11), the pho-toexcitation rate in the plasma with photons emitted viabremsstrahlung is given by λ B γ ( T e , n e ) = 2 π k A i → d γ (cid:90) φ B γ ( E n , E e , T e , n e ) dE e , (12)where the same approximation as for the blackbody spec-trum has been used to solve the integration over the ki-netic electron energy E . The photon flux on resonanceoccurring in Eq. (12) is determined by φ B γ ( E n , E e , T e , n e ) = (cid:18) dσ B ( E e ) dE (cid:19) E = E n φ e ( E e , T e , n e ) . (13)Replacing λ neec by the corresponding photoexcitationrate [Eq. (10) or (12)] in Eq. (7), the total number ofexcited nuclei via resonant nuclear photoexcitation canbe evaluated. A comparison of the NEEC and resonantphoton absorption rates in the plasma is presented inSection III C for a variety of plasma conditions. D. Plasma model
For the plasma modeling part here and in the follow-ing, SI units with k B = 1 are adopted, unless for somequantities the units are explicitely given.
1. General model for spherical plasmas
In order to get a general idea of the number of excitednuclei in the plasma, we first disregard the exact targetheating processes and assume in a first approximation aspherical plasma with homogeneous electron temperature T e and density n e over the plasma lifetime τ p . With thatthe total number of excited nuclei determined in Eq. (7)evaluates to N exc = N iso λ neec ( T e , n e ) τ p , (14)with N iso being the number of isomers in the plasma. N iso can be estimated introducing the isomer fractionembedded in the original solid-state target f iso , N iso = f iso n i V p , (15)where n i stands for the ion number density in the plasmaand the plasma volume V p is given by V p = πR withthe plasma radius R p . In neutral plasmas, the ion andelectron densities are related via the average charge state¯ Z , n i = n e / ¯ Z . (16)In the case of
Mo isomer triggering, an isomer frac-tion of f iso ≈ − embedded in solid-state Niobium foilscan be generated by intense ( ≥ protons/s) beams[24] via the Nb(p,n)
Mo reaction [44].Moreover, the plasma lifetime τ p occurring in Eq. (14)can be approximated for spherical plasmas by followingan estimate for spherical clusters [45]. The time scaleafter which the plasma’s spatial dimension is approxima-tively doubled is given as a function of plasma radius,electron temperature and average charge state by τ p = R p (cid:113) m i / ( T e ¯ Z ) , (17)with the ion mass m i . Note that τ p is implicitly also in-fluenced by the electron density n e due to the dependence¯ Z ( T e , n e ).Based on the expression of the plasma lifetime τ p inEq. (17), the total number of excited nuclei in the plasmacan be estimated. This approximative approach is eas-ily applicable to other nuclear transitions and providesmany instructive insights to plasma-mediated nuclear ex-citations as shown later on considering the example of Mo triggering. In order to test the validity of theplasma lifetime approach, we perform a comparison withresults from a hydrodynamic model for the plasma ex-pansion.
2. Hydrodynamic expansion
Following the analysis in Ref. [25], we consider a moredetailed hydrodynamic model for the plasma expansionby a quasi-neutral expansion of spherical clusters as stud-ied in the context of the intense optical laser pulses in-teraction with spherical clusters [45, 46]. During the ex-pansion, the plasma is assumed to maintain a uniform(but decreasing) density throughout the plasma spherewhile the electron temperature decreases with the adia-batic expansion of the plasma,32 n e,t V dT e,t = − P e dV, (18)where n e,t is the number density of free electrons, V =4 πR / R t ,and P e = n e,t T e,t is the pressure of free electrons. Thetime-dependent electron temperature and the plasma ra-dius satisfy the relation T e,t = T e (cid:18) R p R t (cid:19) , (19)where T e is the initial electron temperature and R p theinitial plasma radius. During the plasma expansion, theelectrons lose their thermal energy to the ions resultinginto the electron and ion kinetic energies n i,t dT i,t dt = − n e,t dT e,t dt , (20)12 m i (cid:18) dR t dt (cid:19) = 32 T i,t , (21)where n i,t is the ion number density, T i,t is the ion tem-perature, and m i is the ion mass. The electron and ioncollisions take place on a much shorter time scale thanthe plasma expansion time, such that we can considerthe temperature to be uniform throughout the sphere[45]. The equation of plasma expansion is given by m i d R t dt = 3 Z T e R R , (22)where Z is the ratio of the electron density to the iondensity, i.e., the average charge state of the ions in thequasi-neutral limit. Solving Eq. (22) for a fixed Z = ¯ Z under the condition that the initial speed for cluster ex-pansion is zero, one obtains the plasma lifetime expres-sion τ p in Eq. (17) as the expansion time for increasingthe plasma radius by a factor 2.Rate estimates based on the FLYCHK code [47, 48]show that the time scale to reach the steady state variesfrom the order of 10 fs for solid-state density to the orderof 10 ps for low density ∼ cm − at temperature ∼
3. Laser-induced plasma: scaling law
Considering the case of a low density (underdense)plasma, which can be generated via the interaction ofa strong optical laser with a thin target, the plasma gen-eration process typically evolves in two steps [49]: (i)a preplasma is formed by the prepulse of the laser; (ii)this preplasma is subsequently heated by the main laserpulse potentially up to keV electron energies. We modelthe plasma following the approach in Refs. [49, 50]. Withthe help of a so-called scaling law, the plasma conditionscan be mapped to laser parameters, like laser intensity I laser , wavelength λ laser , pulse duration τ laser and pulseenergy E pulse . Assuming a flat-top beam profile and fora fixed focal radius R focal the laser intensity reads I laser = E laser τ laser πR . (23)We adopt here at first the widely used ponderomotivescaling law in the non-relativistic limit (sharp-edged pro-files), T e ≈ . I λ µ keV , (24)where I is the laser intensity in units of 10 W/cm and λ µ the wavelength in microns [51–53].Depending on the target and laser-target interactionconditions, different electron temperature scalings areused in the literature [53]. For comparison we adopt herealso a second scaling law (known as short-scale lengthprofile) [53, 54]: T e ≈ (cid:0) I λ µ (cid:1) / keV . (25)The electron density can be estimated as n e = N e /V p where N e is the total number of electrons which can berelated to the absorbed laser energy f E pulse via N e = f E pulse T e . (26)The absorption coefficient f saturates at around 10-15%for high irradiances and steep density profiles [53]. How-ever, in the case of moderate intensities and intermediatescale lengths (e.g. Iλ = 10 W cm − µ m , L/λ ∼ . f taking valuesup to 70% [53].The plasma volume in the case of a laser-generatedplasma is given by V p = πR d p , (27)where the plasma thickness d p = cτ pulse is determined bythe laser pulse duration τ pulse . In contrast to the purespherical plasma, we consider here a cylindrical geometrywith the transversal length dimension determined by thefocal radius of the laser R focal and the longitudinal lengthscale via the plasma thickness d p .Nevertheless, for the case of focal radius, plasma thick-ness and plasma radius of similar scale, we may againconsider the plasma lifetime given by Eq. (17) derived forthe spherical plasma model. For a lower-limit estimateof the nuclear excitation, we use the smallest length scaleout of R focal and d p to calculate τ p , τ p = R focal (cid:113) m i / (cid:0) T e ¯ Z (cid:1) for R focal < d p ,d p (cid:113) m i / (cid:0) T e ¯ Z (cid:1) for d p ≤ R focal . (28) III. NUMERICAL RESULTS FOR SPHERICALPLASMASA. NEEC results
The net NEEC rate λ neec is a function of the prevailingplasma conditions, e.g. electron temperature T e and den-sity n e . As presented in the previous Section, our modelessentially consists of two separate parts which are com-bined to calculate the NEEC rate in the plasma: (i) themicroscopic NEEC cross sections; (ii) the macroscopicplasma conditions like charge state distribution and elec-tron flux.The microscopic NEEC cross sections are calculatedby employing bound atomic wave functions from a Multi-Configurational-Dirac-Fock method implemented in theGRASP92 package [55] and solutions of the Dirac equa-tion with Z eff = q for the continuum. For both typesof wavefunctions, we do not consider the effects of theplasma temperature and density, which are sufficientlysmall to be neglected in the final result of the nuclear ex-citation. The occurring nuclear matrix elements can berelated to the reduced transition probability B ( E
2) forwhich the calculated value of 3.5 W.u. (Weisskopf units)[56] was used.Numerical results for the individual NEEC cross sec-tions of all considered capture channels are presented asa function of the kinetic electron energy in Fig. 1. Werecall that the capture energy needs to coincide with thenuclear transition energy E n , e.g. 4.85 keV in the caseof Mo triggering. Therefore each peak represents aLorentzian resonance located at E res = E n − E α d where E α d stands for the atomic binding energy of the consid-ered capture channel α d . The width of the Lorentzianprofile is given by the natural width of the nuclear trig-gering state T which is composed of the radiative decay Electron energy [eV] n ee c [ b ] L shellM shellN shellO shell
FIG. 1. Microscopic NEEC cross sections as a function ofthe kinetic electron energy. The considered resonance chan-nels for capture into the L -shell (blue, solid), M -shell (or-ange, dashed), N -shell (green, dash-dotted) and O -shell (red,dotted) are shown together with the corresponding resonanceenergy windows (horizontal bars just above the x-axis). and internal conversion channels leading to approx. 130neV.Fig. 1 shows that NEEC prefers the capture into deeplybound states as the cross section peak values increase fordecreasing resonance energy of the free electrons. Theresonance window for L -shell capture lies between 52 and597 eV, for the M shell between 2118 and 4308 eV, for the N shell between 3320 and 4677 eV, and for the O shellbetween 3874 and 4743 eV as illustrated by the horizon-tal lines in Fig. 1. Hence, there is a gap between approx.600 and 2100 eV without any NEEC resonance channels.Note that NEEC into the K shell is energetically forbid-den for the 4.85 keV transition in Mo.The capture into the 2 p / orbital for ions with initialcharge state q = 36 and an initial electron configura-tion of 1 s s p / leads to the highest NEEC resonancestrength (integrated cross section) of 2 . × − b eV.The corresponding resonance energy is at an electronenergy of 52 eV. Interestingly, for higher charge statesNEEC into the L shell is energetically forbidden becausethe binding energies exceed the nuclear transition energyof 4.85 keV.For the calculation of the net NEEC rate in the plasma,the microscopic NEEC cross sections have to be com-bined with the macroscopic plasma parameters accord-ing to Eqs. (1) and (2). We model the plasma conditionsby a relativistic distribution for the free electrons andthe converged charge state distribution computed withthe radiative-collisional code FLYCHK [48] assuming theplasma to be in its non-local thermodynamical equilib-rium (LTE) steady state. The population kinetics modelimplemented in FLYCHK is based on rate equations in-cluding radiative and collisional processes, Auger decayand electron capture. These rate equations are solvedfor a finite set of atomic levels which consists of groundstates, single excited states ( n ≤ P q ( T e , n e ), a simplifiedmodel is used for the atomic orbital population. For aspecific charge state q , we assume (not necessarily to ouradvantage) that the charged ion is in its ground state ini-tially (before NEEC) and capture of an additional elec-tron occurs in a free orbital.Numerical results for λ neec and the total number of ex-cited isomers N exc are presented in Fig. 2 in parallel withthe corresponding electron fluxes and charge state distri-butions. The calculation of the net NEEC rate involvescharge states from q = 14 up to the bare nucleus ( q = 42)with 333 NEEC capture states in total, composed of 5 L -shell, 168 M -shell, 70 N -shell and 90 O -shell orbitals.The results for the dominant recombination channels intothe L and M atomic shells are presented individually inFig. 2. For the total NEEC rate λ neec , further smallercontributions from the recombination into the N and O shells were also taken into account. For the computationof N exc an arbitrary plasma radius of 40 µ m has beenassumed.Both λ neec and N exc increase with increasing electrondensity n e . In the range n e = 10 cm − to 10 cm − ,our calculations show that the charge state distribution P q is nearly unaffected for high temperatures T e . In thesame time, λ neec is enhanced by a factor of 10 maintain-ing almost the same functional dependence on electrontemperature. This indicates that at low densities theboost in λ neec is (almost) a pure density effect comingfrom the increasing number of free electrons present inthe plasma ( φ e ∝ n e ). Increasing the electron density toeven higher values, the behavior of λ neec and N exc be-comes more involved as the charge distribution P q showsa complex dependence on the plasma conditions n e and T e . Between n e = 10 cm − and 10 cm − we see thatwith increasing n e the atomic shell contributions changesignificantly and λ neec is substantially enhanced.Apart from the available charge states in the plasma,the match between the electron distribution [Fig. 2(i)]and the NEEC resonance conditions (Fig. 1) plays an im-portant role for this behavior. The electron distributionsreach their maxima at an energy E ∼ T e . For energiesbelow this value (e.g. where the resonance energies forcaptures into the L shell are located) more electrons areavailable for lower temperatures. In contrast, the highenergy tail of the electron distribution drops exponen-tially with e − E/T e , and is therefore faster decreasing thelower the temperature. In the case of Mo triggering andtemperatures in the keV range, the energy region E (cid:38) T e ,is in particular important for NEEC into the higher shells M , N and O .As seen from Fig. 1, the best case for NEEC would be to have the maximum of the electron distribution( E ∼ T e ) located at the resonance channels with the high-est cross sections (e.g. capture into the L shell). How-ever, assuming that the ions are always in their groundstates initially, this condition cannot be exactly fulfilled,because lower temperatures also lead to lower chargestates present in the plasma such that the L -shell capturewill be closed. The corresponding electron energy win-dow and charge state range for the L -shell resonances arehighlighted by the grey-shaded areas in Fig. 2.These contradicting requirements for efficient NEECsuggest that there is a temperature T max at which theplasma-mediated NEEC triggering reaches a maximumfor each density value n e . The temperatures T max for N exc , the total and partial shell contributions λ neec aredepicted as a function of the electron density in Fig. 2(iii)and Fig. 2(iv). Naively, one would expect that T max isapproximately the same for N exc and for λ neec . This ishowever only true at high densities starting from 10 cm − . According to our approximation in Eq. (17), thechosen plasma lifetime is T e -dependent. In particularat low electron densities τ p acts as a weighting functionproportional to ( T e ) − / shifting the maximum of N exc tolower temperatures. The optimal plasma conditions forthe total excitation number can thus drastically differfrom the optimal conditions for λ neec in this model. Wenote that the arbitrary choice of R p only influences theabsolute scale of the NEEC excitation number, not theposition of T max . B. Ionization potential depression
While the effect of plasma-induced ionization poten-tial depression (IPD) is taken into account for the chargestate distribution (included in FLYCHK), it is neglectedin the calculation of the microscopic NEEC cross sectionsand hence in the NEEC resonance energies so far. In or-der to quantify the effect of the variation of atomic or-bital energy on our final results, we adopted the model ofStewart and Pyatt [57] under the following assumptions:(i) the bound electronic wave functions are unchanged;(ii) the binding energies of atomic orbitals are lowereddue the ionization potential depression, ∆ V ( q, T e , n e );(iii) the free-electron wavefunctions are computed with Z eff = q ; (iv) the kinetic energy of the electrons requiredto match the NEEC resonance condition for a given or-bital is modified according to the reduction of the corre-sponding binding energy. Note that due to our approachfor the potential lowering there might appear additionalNEEC capture channels at low resonance energies (e.g. L shell orbitals), while resonances disappear at resonanceenergies close to the nuclear transition energy.The IPD given by the model of Stewart and Pyatt [57]is (using Gaussian units with k B = 1)∆ V = ze λ D (29) Electron energy [keV] n e =10 cm n e =10 cm n e =10 cm E l e c t r o n f l u x e [ c m s e V ] n e =10 cm n e =10 cm n e =10 cm (i) T e = T e = T e = T e =
20 30 40
Charge state q n e =10 cm n e =10 cm n e =10 cm C h a r g e s t a t e d i s t r i b u t i o n P q n e =10 cm n e =10 cm n e =10 cm (ii) T e = T e = T e = T e = Temperature [keV] × n e =10 cm ×10 × × × n e =10 cm ×10 × × × n e =10 cm ×10 × × N EE C r a t e [ s ] × n e =10 cm ×10 E x c i t a t i o n nu m b e r N e x c × × n e =10 cm n e =10 cm neecLneec Mneec N exc (iii) T max [keV] E l e c t r o n d e n s i t y [ c m ] (iv) neecLneec Mneec N exc FIG. 2. From left to right: (i) Electron flux φ e and (ii) charge state distributions (calculated with the help of FLYCHK) areshown for temperatures T e = 1 keV (blue, solid curve for φ e , blue, solid curve with crosses for charge state distribution), 3 keV(orange, dashed curve for φ e , orange, dashed curve with filled circles for charge state distribution), 5 keV (green, dash-dottedcurve for φ e , green, dash-dotted curve with filled squares for charge state distribution) and 7 keV (red, dash-dot-dotted curvefor φ e , red, dashed curve with filled triangles for charge state distribution) at selected electron densities ranging from 10 upto 10 cm − . Electron energy and charge state ranges for L -shell capture are shaded in grey. (iii) NEEC rate λ neec (blue, solidcurve) and the total number of excited isomers N exc (red, dash-dot-dotted curve), as well as the individual contributions λ Lneec (orange, dashed curve) and λ Mneec (green, dash-dotted curve) from the L and M shell, respectively, are shown as a function of theelectron temperature T e for the selected electron densities n e . A plasma radius of 40 µ m has been assumed in the calculationsof N exc . (iv) The temperatures T max as function of density, for maximizing N exc , λ neec , λ Lneec and λ Mneec , respectively, at eachparticular n e . for the weak-coupling limit, i.e., ( a/λ D ) (cid:28) V = 32 ze a (30)for the strong-coupling limit, i.e., ( a/λ D ) (cid:29)
1. Here, λ D is the Debye length,1 λ = 4 πe T ( z ∗ + 1) n e , (31)and a is the ion-sphere radius for an ion of net charge z defined by 4 πa / z/n e . (32)The net charge z is the charge state of the concernedion (central ion) after the ionization, i.e., the above IPDexpressions (29)-(32) describe the process of removing anelectron from an ion with charge ( z − + . Furthermore, z ∗ =
Temperature [eV] I P D (a) n e = 10 cm n e = 10 cm n e = 10 cm n e = 10 cm Density [cm ] I P D (b) T e = T e = T e = T e = FIG. 3. (a): the relative change in the NEEC rate due to IPDas a function of electron temperature for n e = 10 cm − (blue, solid curve), 10 cm − (orange, dashed curve), 10 cm − (green, dash-dotted curve) and 10 cm − (red, dash-dot-dotted curve). (b): ∆ IPD as function of electron densityfor T e = 1 keV (blue, solid curve), 3 keV (orange, dashedcurve), 5 keV (green, dash-dotted curve) and 7 keV (red, dash-dot-dotted curve). C. Resonant nuclear photoexcitation
Nuclear excitation in the plasma may occur not onlyvia NEEC but also via other mechanisms. In the consid-ered temperature and density range the resonant nuclearphotoexcitation is expected to be the main competingprocess to NEEC. For this reason we evaluated the pho-toexcitation rate in the plasma for two scenarios: (i) con-sidering a black-body radiation spectrum; (ii) consideringa photon distribution originating from bremsstrahlung.The theoretical expressions for the calculation have beengiven in Section II C.The photon flux for TDE conditions (blackbody radi-ation) is presented in Fig. 4. In contrast to the electronflux, φ TDE γ is independent of the electron density anddrastically rises with growing temperature T e . For the Mo isomer triggering especially the flux at 4.85 keVphoton energy is interesting. The flux value increasesfrom 7 × to 9 × cm − s − eV − by going from T e = 1 keV to T e = 7 keV.In Fig. 5, the NEEC rate λ neec is plotted together withthe photoexcitation rates λ TDE γ and λ B γ for electron den-sities 10 , 10 and 10 cm − . A comparison of the Photon energy [eV] [ c m s e V ] T e = T e = T e = T e = FIG. 4. The blackbody photon flux φ TDE γ as a function of pho-ton energy for temperatures T e = 1 keV (blue, solid curve), 3keV (orange, dashed curve), 5 keV (green, dash-dotted curve)and 7 keV (red, dash-dot-dotted curve). NEEC rate and the nuclear photoexcitation assuming ablack-body radiation spectrum at the given plasma tem-perature T e shows that at n e = 10 cm − NEEC domi-nates for T e (cid:46) n e = 10 cm − up to a temperature of 6 keV. We note that whileour NEEC values are to be considered as lower limit es-timates, the actual photoexcitation in the plasma shouldbe lower than the calculated values for a black-body spec-trum in particular at low densities because photons mayeasier escape the finite plasma volume. For the high den-sity n e ≥ cm − parameter regime, NEEC is thedominant nuclear excitation mechanism. The photoexci-tation rate λ B γ was calculated employing bremsstrahlungcross sections dσ B ( E e ) /dE from Ref. [61]. Our resultsshow that the nuclear photoexcitation rate induced bybremsstrahlung photons is always several orders of mag-nitude lower and can be safely neglected. D. Plasma expansion: lifetime approach &hydrodynamic model
So far we considered the lifetime approach to estimatethe total number of excited nuclei via Eq. (14). For amore sophisticated ansatz, we apply the hydrodynamicmodel for the plasma expansion as introduced in SectionII D 2 with initial conditions given by the present plasmaconditions n i , T e and R p . In Fig. 6, the time evolution of λ neec during the expansion is presented for several initialplasma parameters. As seen from the figure, the NEECrate decreases over time since the plasma cools down anddilutes in terms of density while expanding. However,for cases where the initial temperature exceeds T max forthe given initial density, the NEEC rate first increases,peaking out at optimal conditions and afterwards followsthe typical decaying pattern, as clearly visible in Fig. 6.A timescale comparison between the hydrodynamicmodel and the lifetime approach according to Eq. (17)0 Temperature [eV] E x c i t a t i o n r a t e [ s ] neec : n e = 10 cm neec : n e = 10 cm neec : n e = 10 cm TDEB : n e = 10 cm B : n e = 10 cm B : n e = 10 cm FIG. 5. NEEC rate λ neec [blue (dark gray)] and photoexcita-tion rates λ TDE γ [orange (light gray)] and λ B γ [green (mediumgray)] as functions of electron temperature. For NEEC andfor photoexcitation via bremsstrahlung photons, electron den-sities of 10 cm − (solid lines), 10 cm − (dashed lines) and10 cm − (dash-dotted lines) have been considered. Time [ps] n ee c [ s ] × n e = 10 cm (a) Time [ps] n ee c [ s ] × n e = 10 cm (b) Time [ps] n ee c [ s ] n e = 10 cm (c) T e = 1 keV T e = 3 keV T e = 7 keV FIG. 6. NEEC rate λ neec as a function of time during thehydrodynamic expansion. We considered initial electron den-sities 10 cm − (a), 10 cm − (b) and 10 cm − (c) andinitial temperatures of 1 keV (blue curves with filled circles),3 keV (orange curves with filled squares) and 7 keV (greencurves with crosses). The lifetime estimate is illustrated bythe dashed lines. (illustrated by the dashed lines in Fig. 6) shows thatthe latter seems to overestimate the NEEC timescale es-pecially for low temperatures. Also, the dependence ofthe NEEC timescale on T e is much weaker for the hy-drodynamic expansion as determined by Eq. (17). Forinstance, for T e = 1 keV and T e = 7 keV at densities n e = 10 cm − , the lifetime τ p is given by 226 and 77ps, respectively. Considering the hydrodynamic expan-sion, the time integration over λ neec roughly convergesafter 110 and 130 ps, respectively.Despite this discrepancy in the NEEC timescales be-tween the lifetime and hydrodynamic models, the com-parison of the total number of excited nuclei as a function Temperature [eV] N e x c (a) n e = 10 cm n e = 10 cm n e = 10 cm p Hydro Density [cm ] N e x c (b) T e = 1 keV T e = 3 keV p Hydro
FIG. 7. NEEC excitation number as a function of tempera-ture (a) and electron density (b). Results from hydrodynamicexpansion (solid curves) are shown in comparison to the life-time estimate τ p (dashed curves) for several initial plasmaconditions. An initial plasma radius of 40 µ m has been con-sidered. of the plasma conditions in Fig. 7 shows a strickingly sim-ilar behaviour for the two models. For the calculationsa plasma radius of 40 µ m has been used. As a rule, theexcitation numbers for the hydrodynamic expansion areslightly smaller than the corresponding ones from the life-time approach. Furthermore, the highest deviation be-tween the expansion models is at low temperatures andsmall densities as already expected from the results inFig. 6. At a temperature of 1 keV and an electron den-sity of 10 cm − the relative difference in the estimatedexcitation numbers N exc evaluates to 84%, while it is onthe order of 5% at the high temperature ( ≥ cm − ) tail.Overall, the lifetime approach appears to provide rea-sonable estimates for N exc which deviate from the hy-drodynamic model by 40-80% for low temperatures andsmall densities. At high T e and n e , the predicted exci-tation numbers almost coincide. As an advantage, theplasma lifetime approach is less computationally expen-sive and can be applied to a broader range of problemsin comparison to the hydrodynamic expansion. For anorder of magnitude estimate we therefore proceed to ex-tract the optimal NEEC parameters for optical-laser gen-erated plasmas with the help of the plasma lifetime τ p approach.We note that, following the argument in Ref. [45], weassume for Eq. (21) that all ions have the same velocity dR t /dt during the expansion. If one assumes the ionvelocity at position r to be dr/dt and the velocity scales1linearly with the position, the factor 3 in Eq. (22) shouldbe replaced by 5 [62, 63]. However, the difference betweenthese two factors should not affect our conclusion on thevalidity of the lifetime approach for the nuclear excitationcalculation. Finally, the expansion with uniform densityadopted above is a rather simplified model but it providesgood estimates of the cluster expansion characteristics.For a more accurate model, however beyond the scopeof the present work, we refer the interested reader toRef. [64]. IV. LASER PLASMAS
In the following we proceed to determine how the opti-mal NEEC parameter region in the temperature-densitylandscape may be accessed by a short laser pulse. Wediscern in our treatment two cases, namely the low- andhigh-density plasmas, and refine accordingly our plasmamodel.
A. Low density
First, we consider the case of a low-density (under-dense) plasma, which can be generated via the interac-tion of a strong optical laser with a thin target. Theplasma generation process typically evolves in two steps[49]: (i) a preplasma is formed by the prepulse of thelaser; (ii) this preplasma is subsequently heated by themain laser pulse potentially up to keV electron energies.We model the plasma following the approach presentedin Section II D 3 with the help of so-called scaling lawswhich provide a unique relation between laser parametersand plasma conditions. We employ two scaling law mod-els, the sharp-edge scaling law (ponderomotive scalinglaw) in Eq. (24) further denoted as SL1, and the short-scale length profile scaling law in Eq. (25) referred to inthe following as SL2.
1. Results of scaling laws
In Fig. 8(a) and Fig. 8(b) the electron temperature T e and the number of free electrons N e in the plasma areshown as functions of the laser irradiance Iλ , respec-tively, for both scaling laws SL1 and SL2. The consid-ered range of irradiances has been chosen such that theexpected electron temperatures span from approx. 300eV to 8 keV.SL1 and SL2 should be valid for collisionless heating inthe intermediate irradiance regime Iλ ∼ W cm − µ m [53]. In particular, SL1 covers the entire param-eter regime presented in Fig. 8, while SL2 exceeds itsvalidity domain for the low-temperature case. For thesake of comparison, in the following we compare SL1 andSL2 throughout the entire parameter regime of interest. However, we should keep in mind that at low tempera-tures (lower than ∼ Iλ ∼ W cm − µ m ) [53, 54], in which the collisional heating may be thedominating heating mechanism.In order to estimate the number of free electrons inthe laser-heated plasma the laser absorption coefficient f occuring in Eq. (26) needs to be fixed. In Refs. [65, 66] ex-perimental data on the absorption of short laser pulses inthe ultrarelativistic regime are presented. A Ti:sapphirelaser with 150 fs pulses at 800 nm and an energy up to20 J was used to heat Al foils (thickness ∼ . − µ m)and Si plates (thickness ∼ µ m). The measured laserabsorption shows no significant dependence on the tar-get thickness and the material. Moreover, in consistencywith previous experiments at lower intensities [66], itcould be shown that the absorption mechanisms changefrom collision dominated to collisionless by exceeding anintensity of around 10 W/cm . The experimental re-sults show a good agreement with a theoretical calcula-tion based on a Vlasov-Fokker-Planck code. We thereforeadopt a universal absorption coefficient f = f ( Iλ ) as afunction of laser irradiance to estimate the absorbed laserenergy by peforming a cubic interpolation to theoreticalresults based on a Vlasov-Fokker-Planck code presentedin Ref. [65]. A more detailed discussion on the laser ab-sorption coefficient and its impact on the NEEC rates isgiven in Section IV A 2.As can be seen from Fig. 8, SL2 predicts higher tem-peratures for a given laser intensity I laser . However, sincethe number of free electrons is inversely proportional tothe electron temperature [see Eq. (26)], the resulting den-sity for SL2 is expected to be smaller in comparison toSL1. For a wavelength of λ laser = 1053 nm typical forNd:glass lasers and a pulse energy of 100 J, the corre-sponding temperature-density profiles for SL1 and SL2are presented in Fig. 8(c). With the considered intensityrange between 7 × and 2 × W/cm for SL1,the absorption fraction f lies between 0.1 and 0.2 lead-ing to electron densities in the order of 10 cm − . Theextension of the low intensity tail for SL2 (10 − W/cm ) results in slightly higher absorption coefficientsup to 30%. However, the electron densities are smaller inthis case since the average electron temperature is higherfor a fixed plasma volume V p .Numerical results for λ neec and for the total excitationnumber N exc per laser pulse are presented in Fig. 9 asfunctions of the laser intensity. The plasma expansiontime is estimated by using the lifetime approach with thesmallest length scale out of R focal and d p [see Eq. (28)]for a lower-limit estimate of the NEEC excitation. Weconsider a pulse energy of 100 J, wavelength of 1053 nm,and laser pulse duration values of 500 fs. Apart from λ neec and N exc , the number of isomers N iso present inthe plasma, the average charge state ¯ Z and the plasmalifetime τ p in units of τ laser are shown as functions of thelaser intensity.Analog to the discrepancy between T max for NEEC2 I [W cm m ] T e [ e V ] (a) SL 1SL 2 I [W cm m ] N e E laser = 100 J (b) T e [eV] n e [ c m ] E laser = 100 J laser = 1053 nm (c) FIG. 8. Accessible plasma parameters for ponderomotive scaling law [SL1, blue (dark gray) curves] and short-scale lengthprofile scaling law [SL2, orange (light gray) curves]. The electron temperature (a) and the number of free electrons presentin the plasma (b) are presented as functions of the laser irradiance Iλ . In the calculations of N e a pulse energy of 100 J isassumed. The graph (c) shows the corresponding plasma profile n e = f ( T e ) for a laser wavelength of 1053 nm, a pulse energyof 100 J and intensity ranges 7 × − × W/cm (SL1) and 10 − W/cm (SL2). The results here are independentof τ laser . I [W/cm ] N i s o (i) I [W/cm ] Z (ii) I [W/cm ] p / l a s e r × (iii)10 I [W/cm ] n ee c [ s ] (a) SL 1SL 2 I [W/cm ] N e x c (b) FIG. 9. The NEEC rate λ neec (a) and the total excitationnumber N exc per laser pulse (b) as functions of laser intensityfor ponderomotive scaling law [SL1, blue (dark gray) curves]and short-scale length profile scaling law [SL2, orange (lightgray) curves]. Upper graphs, from left to right: (i) the numberof isomers present in the plasma, (ii) the average charge state¯ Z and (iii) the plasma lifetime are presented. See text forfurther explanations. rate and total excitation number, also here the optimallaser intensities I opt at which λ neec and respectively N exc are maximal do not coincide. For the assumed laserparameters, λ neec is maximized by I opt = 1 . × W/cm at a temperature of 5 keV and a density of5 . × cm − in the case of SL1. In contrast, theoptimal intensity for N exc per laser pulse is 3 . × W/cm . The electron temperature and density achievedat this intensity are 1.4 keV and 7 . × cm − , respec-tively, leading to a charge state distribution with ¯ Z ∼ M shell still dominatethe L -shell contribution.The optimal values for SL2 lie at smaller intensities,3 . × W/cm and 1 . × W/cm for λ neec and N exc , respectively, where plasma conditions T e = 6 keV, n e = 3 . × cm − and T e = 2 . n e = 4 . × cm − are prevailing.In general, Fig. 9 shows that the total number of ex-cited isomers is maximal at plasma conditions with loweraverage charge state ¯ Z but longer plasma lifetime andlarger plasma volume in comparison to the optimal con-ditions for the NEEC rate. The effect of the larger plasmavolume can be nicely seen for the number of isomerspresent in the plasma [Fig. 9(i)] which is given by ap-proximatively 2 × isomers at I opt for N exc . The pro-portionality of N iso with respect to the laser irradiance Iλ is given by the following relation, N iso ∝ ¯ ZN e ∝ (cid:40) ¯ Zf ( Iλ ) / ( Iλ ) for SL1¯ Zf ( Iλ ) / ( Iλ ) / for SL2 , (34)where ¯ Z is itself a function of laser intensity and wave-length for fixed pulse energy E pulse .
2. Laser absorption
Since the laser absorption fraction f can be treatedas free parameter in the scaling laws SL1 and SL2, westudy the effect of different f values for the NEEC ratein the plasma and the corresponding nuclear excitation.We have performed calculations with constant absorptionfractions 10%, 20% and 30% considering a pulse energy of100 J, a laser wavelength of 1053 nm and a pulse durationof 500 fs. The results together with a comparison withthe laser absorption model f ( Iλ ) are shown in Fig. 10for SL1 and Fig. 11 for SL2.As can be seen from these two figures, the NEEC rateas well as the total excitation number increase with in-creasing absorption coefficient f since higher densities arereached. Moreover, the optimal laser intensities dependon the laser absorption as illustrated in the correspond-ing insets of the graphs. While the change of I opt due to3 n ee c [ s ] (a) f ( I ) f = f = f = I [W/cm ] N e x c (b) I opt [ W/cm ]02 n ee c [ s ] ×10 I opt [ W/cm ]0.02.5 N e x c ×10 FIG. 10. Impact of the laser absorption f on NEEC resultsconsidering SL1. λ neec (a) and N exc (b) are presented as func-tions of I laser for f = f ( Iλ ) (blue, solid line), f = 0 . f = 0 . f = 0 . f values. See text forfurther explanations. n ee c [ s ] (a) f ( I ) f = f = f = I [W/cm ] N e x c (b) I opt [ W/cm ]02 N e x c ×10 FIG. 11. Impact of the laser absorption f on NEEC resultsconsidering SL2. Analog notation as in Fig. 10. f is smaller than 10% and hence negligible in terms ofthe expected accuracy of our model for SL1, the intensityand wavelength-dependent absorption coefficient f has amuch stronger effect on the predictions of SL2 in com-parison to a constant absorption fraction. With constant f the optimal intensity for N exc is at approx. 2 . × W/cm in comparison to 1 . × W/cm with f ( Iλ ).Note that I opt for λ neec is not shown in Fig. 11 since the
600 800 1000 1200 1400 [nm] I [ W / c m ] (a) N e x c
200 400 600 800 1000
E [J] I [ W / c m ] (b) N e x c FIG. 12. Total number of excited isomers as a function oflaser parameters. N exc ( I, λ ) for fixed E pulse = 100 J (a) and N exc ( I, E ) for fixed λ laser = 1053 nm (b) are presented. Thepulse duration is assumed to be τ laser = 500 fs. rate keeps increasing for higher intensities way out of thevalidity range for the applied scaling law model SL2.
3. Dependence on laser parameters
In this Section, we analyze the functional behavior ofthe NEEC excitation on the laser parameters I laser , λ laser , τ laser and E pulse . For this analysis we restrict ourselvesto the steep density gradients scenario which is best de-scribed by SL1 using the universal absorption coefficient f ( Iλ ). In Fig. 12(a) we present the total number ofexcited isomers N exc as a function of laser intensity andwavelength for fixed laser pulse duration 500 fs and pulseenergy 100 J. It can be seen that the highest excitationnumbers can be found for small wavelengths at the cor-responding optimal intensity I opt illustrated by the redcrosses for given λ laser . The optimal intensity values areincreasing with decreasing laser wavelength. We recallthat smaller wavelengths lead to smaller electron densi-ties which require a higher temperature to maximize theNEEC excitation (cf. Fig. 2).However, typically the wavelength is a parameter de-termined by the fundamental laser design (i.e. 1053 nm4in the case of Nd:glass lasers, or 800 nm for Ti:sapphirelasers) and can only be changed by considering higherharmonics. Therefore it is worth to further investigatethe behavior of N exc in terms of variable laser intensityand pulse energy for a given wavelength of, for instance1053 nm, and pulse duration of 500 fs. Numerical resultsare shown in Fig. 12(b). As can be expected alreadyfrom the direct relation between number of free electronsin the plasma and E pulse [compare Eq. (26)], the NEECexcitation is higher for higher pulse energies. The opti-mal laser intensity for given E pulse [represented by thered crosses in Fig. 12(b)] is constant over the consideredenergy range.For d p < R focal (the case for the parameters of Figs. 9- 12) the plasma lifetime is determined by d p and in turnby τ pulse . In order to evaluate the influence of τ pulse onthe NEEC excitation, Fig. 13 shows N exc at the optimalintensity I opt as a function of the pulse duration. Forthe calculations we considered again a fixed pulse energyof 100 J and a wavelength of 1053 nm. As seen fromthe figure, the NEEC excitation becomes stronger withincreasing laser pulse duration τ pulse reaching its maxi-mum at 2.2 ps, the value where d p = R focal . Accordingto our model, this condition is satisfied for τ laser = (cid:18) E laser c πI laser (cid:19) / . (35)For even longer pulse durations we need to use R focal inour model to determine the plasma lifetime [see Eq. (28)].We then notice a decrease of N exc as for a given laserpulse energy, longer τ pulse values require smaller focalradii to obtain the same intensity and in turn shorterplasma lifetime. The optimal intensity shifts slightly tosmaller values by going from the parameter region where d p < R focal to parameters with d p > R focal (see inset ofFig. 13).We note that for short-pulse lasers, the plasma lifetimeis constrained by the plasma thickness which depends onthe pulse duration and optimally should have a size simi-lar to the other two dimensions encompassed by the focalspot. We have not considered here long nanosecond laserpulses which lead to a complex plasma evolution whichis difficult to model. While short laser pulses limit theplasma lifetime, long, ns laser pulses in turn come witha small focal spot to obtain the necessary laser intensi-ties. Effectively, the nuclear excitation should be similarin magnitude at a given pulse energy for both short-pulseand ns-pulse lasers.
4. High-power laser facilities
In this Section, we evaluate the optimal laser intensity I opt and the expected maximal NEEC excitation N exc forrealistic parameters of high-power optical lasers whichare available currently or under construction. Resultsfor the ELI-beamlines L4 [67], ELI-NP [68, 69], PETAL laser [ps] N e x c × laser [ps] I o p t [ W / c m ] × FIG. 13. Total excitation number N exc at optimal intensityas a function of τ laser . The inset shows I opt ( τ laser ) close tothe region where N exc is maximized. We considered a pulseenergy of 100 J and laser wavelength of 1053 nm. [70], LULI [71], VULCAN [72] and PHELIX [73] lasersare presented in Table I.For all considered cases, the excitation N exc per laserpulse is orders of magnitude larger than the one in theXFEL-generated cold plasma [24, 25]. We note that inthe analysis in Refs. [24, 25], a B ( E
2) value of 1 W.u.and the isomer fraction of ∼ . × − were used. Forthe comparison presented here, we have recalculated theXFEL excitation number using B ( E
2) = 3 . ∼ − ) considered for the presentwork yielding the result N exc ∼ − for a T e = 350 eVplasma. Moreover, the largest value of 2 . τ pulse . Table I and Figs. 12 and 13 show that a balancebetween the laser power and laser pulse duration is ben-eficial for the excitation number.Note that the values presented here slightly differ fromthe values provided in Ref. [28], since we took into ac-count an additional data point for the fitting of the uni-versal absorption coefficient f to extend the model tosmaller irradiances Iλ . B. High density
We now turn to the case of high electron densities,which promises the strongest nuclear excitation accord-ing to Fig. 2. Experiments and simulations have shownthat it is possible to isochorically heat targets at solid-state density to temperatures of a few hundred eV or evena few keV [38, 74, 75]. Since in this regime the heatingof the target is mainly conducted by secondary particles,i.e. hot electrons generated in the laser-target interac-tion, a more sophisticated model is necessary comparedto the low-density case.5
ELI-beamlines ELI-NP PETAL LULI VULCAN PHELIX E pulse [J] 1500 250 3500 100 500 200 τ pulse [fs] 150 25 5000 1000 500 500 λ [nm] 1053 800 1053 1053 1053 1053 N exc . × − . × − . . × − . × − . × − TABLE I. Laser parameters and maximal N exc achieved at the optimal laser intensity I opt = 3 . × W/cm for ELI-beamlines L4 [67], PETAL [70], LULI [71], VULCAN [72] and PHELIX [73] and I opt = 5 . × W/cm for ELI-NP [68, 69]lasers.
1. PIC simulation
The solid-state isomer target is practically a Niobiumfoil with a 10 − fraction of embedded Mo isomers. Wehave performed a one-dimensional (1D) particle-in-cell(PIC) simulation of a Nb solid target with 1 µ m thicknessand Nb density of n nb = 5 . × cm − interactingwith a high-power laser using the EPOCH code [76]. Theisomer fraction is small enough to be neglected here inthe determination of the plasma conditions. The laserhas a Gaussian profile in time with peak intensity I =10 W/cm , laser duration τ pulse = 500 fs, and laserwavelength λ = 800 nm, respectively. At the boundary ofthe simulation box where the laser is introduced, the laserreaches the peak intensity at time t = 500 fs. A linearpreplasma with the thickness of 0 . µ m is considered infront of the solid target. The simulation box is 4 µ m inlength, and the solid target is placed at the center of thesimulation box. Ionization is not included explicitely inthe simulation; as a representative order for the electrondensity, we fix the charge state to 10.To include the effect of atomic ionization and recom-bination events, we averaged the raw data for electrontemperature T e and ion density n i from the PIC sim-ulation over 10 nm intervals, and used these values asinput for the radiative-collisional model implemented inFLYCHK [48] to obtain charge state distributions and(corrected) electron densities. The electron density andtemperature values are shown in the lower and middlepanels of Fig. 14 for a number of time instants between1.5 and 3.5 ps as a function of the target penetrationdepth x together with first order polynomial and thirdorder exponential fits, respectively. We note that due toionization effects the real temperature is expected to bedifferent from the one obtained in the PIC simulation.We use the latter only as first approximation.
2. NEEC excitation
For the high-density region, we evaluate the NEECrate as a function of target depth x and time t by insert-ing the PIC-simulation results for T e and the corrected n e values into Eqs. (1) and (2). The plasma is assumed tobe homogeneous only in the plane perpendicular to the x direction over the region of A focal . We consider a laser pulse energy of 100 J, which leads for the pulse durationand laser intensity adopted in the PIC simulation to afocal spot area of approximatively 2 × − cm . Resultsfor λ neec and λ TDE γ are presented as a function of x forfive time points between t = 1 . t = 3 . x where optimal plasma conditions are prevailing.The peak propagates through the target and disappearsat around 4 ps as the target heating leads afterwards totemperatures exceeding the optimal value for NEEC. Adetailed analysis of data sampled from 1 to 4 ps in 100-fssteps shows that the integrated NEEC rate reaches itsmaximum at 3.1 ps and drops roughly to half its valueat 4 ps. Due to the high electron density, λ neec is muchlarger than the photoexcitation rate over the entire tar-get.The total NEEC rate is shown together with its indi-vidual L - and M -shell contributions λ Lneec and λ Mneec , re-spectively, in Fig. 15 at the time instant of 3.1 ps wherethe integrated value reaches its maximum. The figureshows that the flat region mainly comes from the cap-ture into the M shell, which is available (almost) overthe whole temperature-density landscape. In contrast,the L -shell NEEC orbitals are only accessible in a verylimited region in terms of plasma conditions, leading tothe peak in λ neec at a target depth x where optimal con-ditions for L -shell NEEC are prevailing.Using the regression curves for λ neec calculated withthe fitted n e and T e functions, we solve Eq. (7) in a two-step procedure to obtain the total NEEC excitation num-ber N exc . First, for each time instant t the product ofthe NEEC rate and the isomer density is integrated withrespect to x over the entire target thickness d t and mul-tiplied by the focal spot area A focal to account for theperpendicular directions. Second, the outcomes of thespatial integration are interpolated as a function of timeleading to N exc ( t ) which is defined as the derivative withrespect to time of the number of excited isomers frominitial time t (here 1 ps) up to t , N exc ( t ) = (cid:90) V p d r n iso ( r , t ) λ neec ( T e , n e ; r , t ) . (36)The interpolation for N exc ( t ) is then inserted in the timeintegral in Eq. (7) which is solved numerically.For t > N exc ( t ) assuming an ex-ponential functional behavior initially following the slope6 [ s ] T e [ k e V ] x [nm] n e [ c m ] x [nm] 2.5 ps x [nm] 3.0 ps Raw PIC neecTDE x [nm] 3.5 ps
Regression neecTDE x [nm]
FIG. 14. Nuclear excitation rate λ (upper row), plasma temperature T e (middle row) and electron density n e (lower row) basedon the PIC simulation as functions of target depth x at time instants 1.5 ps, 2.0 ps, 2.5 ps, 3.0 ps amd 3.5 ps. The NEEC rate λ neec [blue (dark gray)] is shown in the upper-row graphs together with the photoexcitation rate λ TDE γ [orange (light gray)].The laser has peak intensity I = 10 W/cm and wavelength λ = 800 nm. The raw data averaged over 10 nm intervals ispresented together with a linear polynomial and a third order exponential fit for n e and T e , respectively. The raw result of λ neec and regression curves for λ neec calculated with the fitted n e and T e functions are shown in the upper row graphs. x [nm] n ee c [ s ] Time: 3.1 ps neecLneecMneec
FIG. 15. NEEC rate λ neec and regression curves for λ neec calculated with the fitted n e and T e functions are shown forthe total rate [blue (dark gray)] as well as individual L -shell[orange (light gray)] and M -shell [green (medium gray)] con-tributions as a function of target depth x . at 4 ps. The time integration starting from 1 ps con-verges approximatively after 10 ps, leading to an excita-tion number of 1 .
3. Modeling of the plasma expansion
The extrapolation method described above is equiva-lent with the assumption that the plasma heating contin-ues after 4 ps. However, since no further energy is placedinto the system, the plasma heating should reduce andfinally turn into cooling during the plasma expansion.For a cross-check, we consider the plasma expansion toset in directly at 4 ps, and use a hydrodynamic model toestimate N exc . We consider the average ion density andelectron temperature at 4 ps as input for our hydrody-namic expansion model introduced in Section II D 2 as-suming homogeneous plasma conditions over the plasmavolume V p = A focal d t . The results for the excitation num-ber differential in time are shown in Fig. 16 for both theextrapolation as well as the expansion model.During the cooling phase, N exc ( t ) reaches again a lo-cal maximum at the time where T e = T max for the givendensity (see Fig. 16). However, since the density is alsostrongly decreasing during the expansion, the net effectfor the excitation number is small. Our calculations showthat the extrapolation method as well as the hydrody-namic expansion deliver similar results for N exc with adeviation of 10%. Moreover, the values from both meth-ods are in good agreement (within a 20% interval) witha simple lifetime estimate where according to Eq. (17)the plasma lives for additional 2 ps with homogeneousplasma conditions given by the averaged values at t = 4ps.Note that the PIC simulation has been carried out inthe direction with the smallest length scale of the plasma,such that our model underestimates the plasma lifetimeand only gives a lower limit for the excitation number.Modeling the expansion in the perpendicular direction of7 Time [ps] N e x c ( t ) [ s ] × Integrals: N extraexc = N hydroexc = LASER
PICExtrapolationHydrodynamic expansion
FIG. 16. The time-dependent nuclear excitation number N exc ( t ) for PIC results (blue circles). For t > N exc . The laser pulseduration is schematically illustrated by the yellow area. the laser incidence with the length scale set by the focalradius, a 10 to 100-fold longer lifetime can be expectedto boost N exc . V. CONCLUSIONS
Our results show that by a proper choice of target andoptical laser parameters, the plasma conditions can be tailored to optimize nuclear excitation via the NEECprocess. For the case of
Mo, both low-density andhigh-density plasmas promise observable depletion of theisomer. The induced excitation is expected to be six or-ders of magnitude higher than secondary NEEC in anXFEL-produced cold plasma and in turn a factor 10 up to 10 higher than the direct photoexcitation withthe XFEL. Allegedly the absolute number of depletedisomers remains small, mainly due to the fact that thenumber of isomers in the microscopic plasma volume issmall and only a 10 − fraction of them gets depleted.The excitation number of approximatively 2 isomersper pulse from our conservative estimate together withlaser repetition rates of up to tens of Hz for 100 J pulsesreach for the first time the threshold of one isomer de-pletion per second and should provide a detectable sig-nal. The experimental signature of the nuclear excita-tion in the plasma would be a gamma-ray photon ofapprox. 1 MeV released in the decay cascade of thetriggering level in Mo. 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