Coulomb cluster explosion boosted by a quasi-dc pulse -- diagnostic tool and ultimate test of laser fusion efficiency in clusters
aa r X i v : . [ phy s i c s . a t m - c l u s ] M a y Coulomb cluster explosion boosted by a quasi- dc pulse –diagnostic tool and ultimate test of laser fusion efficiency in clusters A. E. Kaplan ∗ Electr. and Comp. Eng. Dept, The Johns Hopkins University, Baltimore, MD 21218 (Dated: May 8, 2018)To greatly enhance output of nuclear fusion produced neutrons in a laser-initiated Coulomb explo-sion of Deuterium clusters, we propose to accelerate the resulting ions by a quasi- dc electrical pulseto the energies where the D + + D collision cross-section is the highest. With D + ions bombardingthen a Deuterium-rich solid-state cathode, this allows one to solve a few problems simultaneously by(a) completely removing electron cloud hindering the Coulomb explosion of ionic core, (b) utilizingup to 100% of the cluster ions to collide with the high-density packed nuclei, and (c) reaching highlyincreased cross-section of neutron production in a single D + + D collision, in particular by using amulti-layered target. We also consider the use of E -pulse acceleration for diagnostic purposes. PACS numbers: 36.40.Gk, 52.50.Jm, 25.45.-z, 79.77.+g
Nuclear fusion reactions in solid-deuterium laser-produced plasma were first observed almost 50 years ago[1]. A promising recent development in the field is genera-tion of neutrons in laser-induced ionization and explosionof deuterium clusters [2] with a conversion efficiency upto ∼ × neutrons/J [3]. A major mechanism here isa Coulomb explosion, CE [2-5] of the clusters, wherebyan irradiated and highly ionized cluster loses its free elec-trons that ideally are almost instantly swept away by thelaser, with the ionic core torn apart by repulsive Coulombforces resulting in CE. Part of the process is the forma-tion of shock-shells in expanding ionic cloud predictedin [6], explored in detail in [7] and most recently experi-mentally observed in [8]. In general, however, there couldbe some other mechanisms of cluster explosion, such ase. g. the quasi-neutral micro-plasma and hydrodynamicmodels [9], etc., more characteristic for lower laser inten-sities ( < W/cm ). While the generation of neutronsindicating nuclear fusion reaction have been successfullyobserved [1-3], the results are still far from the goal ofdecades-long quest for an elusive efficient nuclear fusionfor energy-producing application, whereby the energy ofgenerated neutrons exceeds that of the input.Further advance in using laser-induced explosions ofclusters is blocked by three major problems: insufficientkinetic energy of ions (typically a few KeV , with colli-sional cross-section, σ f , very low), v s ∼ KeV where σ f goes up by orders of magnitude, see below; low utiliza-tion of produced ions due to low density of surroundingplasma (typically < cm − v s ∼ cm − in solidstate), hence low number of fusion collisions; and finally,free electrons that eventually neutralize the ion cloud andhamper Coulomb explosion. Shock-shells in CE [6,8] mayincrease that collision rate [6] by having ions collide inthe higher-density area near original cluster, but this en-hancement is still insufficient to attain good efficiency.In this Letter we propose: (1) the way of overcom-ing those problems by using a laser-synchronized electri-cal pulse to accelerate laser-produced ions D + to ener- gies sufficiently high (up to ∼ − KeV ) to greatlyincrease the cross-section for the neutron production,and smashing them against a deuterium (or tritium)-richsolid-state target (cathode); aside from greatly enhanc-ing neutron output, this would provide an ultimate testof laser+cluster nuclear fusion for energy production ap-plications, (2) the enhancement of the solid-state targetperformance by making it as a stack of thin layers, and(3) a diagnostic of the cluster and explosion structure forresearch purposes and application as a neutron source.This approach may be viewed as a cross between laser-induced CE and basic electrostatic generation of neutrons[12] providing substantial cross benefits. First, there is noneed anymore to strive for powerful laser irradiation, inparticular to remove the electron cloud; the laser energyhas to be just enough to attain a reasonable ionization,not to produce high ion energies or blow electron cloudaway. Ionized electrons here are quickly removed fromthe expanding cloud, thus barring them from neutraliz-ing the ions in the cloud and hindering the useful effectof CE. Similarly, laser-induced ionization may allow touse lower E-field than in electrostatic generators. Thesystem also simplifies the analysis and diagnostics of theprocess: once electrons are removed, the dynamics of re-maining cloud of positive ions is strictly due to a repulsiveCoulomb explosion, which now may differ from an idealCE only in that in each case the radial density distribu-tion of the ions may be non-uniform depending on theirinitial distribution. The latter one can thus be elicited byusing segmentation of the cathode into a few electricallyisolated sections/rings and recording the time-dependantcurrent from each one of them, see below.In a common arrangement, a cluster or a jet of clustersis injected between an anode collecting ionization pro-duced electrons and a cathode covered by deuterium ortritium-rich material, as a target for D + ions acceleratedby a strong E -pulse applied to the electrodes. The ion en-ergy then can get much higher than that produced by CEand reach the optimum domain of up to 100 KeV ; it is di-rectly controlled by the E -pulse amplitude. A solid-statetarget insures then a high probability of fusion collisions.An E -pulse has to be sufficiently long to be maintainedtill all the ions reach the cathode. Its duration for e.g. 50 KeV voltage and 2 cm electrode spacing, has to be > ns , while the laser pulse is typically sub- ps long; theformation time of ion cloud is even much shorter. Fora laser intensity ≫ W/cm , free electrons are pulledfrom the core faster than a laser cycle [6], and then theyare swept away by the E -pulse and brought to the anode.Electrode geometry may vary from spheres to cylin-ders, and to cones, while a parallel configuration providesfor the simplest arrangement, Fig.1. Consider a cluster F u ll i on f l o w r a t e , ( d N / d t ) / N S Dimensionless time, t =t / t c =2.5 c =6 c =12.5 c =25 0 0.5 1 -0.4 0 0.4 D i s t an c e f r o m c a t hode , z / z (Cross-section distance)/z clusterexplosionioncloud c =25 t =0.5 t =1 ✲ cathode 0 0.5 1 -0.4 0 0.4 D i s t an c e f r o m c a t hode , z / z (Cross-section distance)/z clusterexplosionioncloud c =25 t =0.5 t =1 ✲ cathode 0 0.5 1 -0.4 0 0.4 D i s t an c e f r o m c a t hode , z / z (Cross-section distance)/z clusterexplosionioncloud c =25 t =0.5 t =1 ✲ cathode FIG. 1: Full ion flow rate, ( dN/dτ ) / N Σ vs dimensionlesstime, τ = t/t , for various potential/kinetic energy ratios, χ = U /T max . Inset: the schematics of the proposed experi-ment; dotted circles outline the edge of ion cloud after clusterexplosion at two different moments. The expanding positivelycharged cloud moves down toward the deuterium-rich cathodeand finally hit it under the action of high-voltage potential. initially at the distance z from a cathode and a potentialbetween them is U = qV , where V is a respective volt-age and q – an ionic charge (for D + , q = e ). Once boundelectrons are freed by laser pulse, the E -pulse “vacuum-cleaning” pulls them to the anode. The ensuing ionic CEwould then typically produce a shock at the edges of aCE cloud [6-8], yet soon the density of over-run ions de-creases. Then, for an ideal CE, the expansion proceeds asif a sphere with uniform density distribution in the mo-mentum space. In general, however, this could be morecomplicated, due to expansion originated by thermal ex-plosion, secondary ionization, or compositions, such ase. g. non-uniform or heterogeneous clusters [3,13] ofdifferent ionic species, or mixed clusters formed by de-positing layers of different atoms, etc. This may resultin distinctly different initial non-uniform ion kinetic en-ergy radial distribution, with the same maximum energy T max , see below. In the case of an ideal CE, wherebythe initial density is almost uniform, we have [6] T max = ( en i ) N Σ /R , (1) where N Σ is a total number of ions in a cluster, n i e is theion charge (for hydrogen atoms or isotopes, n i = 1), and R is an original radius of cluster. When R cl >> R ,where R cl ( t ) is a cloud radius, the ion motion is unaf-fected by ion collisions, and the ion movement is iner-tial in the frame of the center of mass (COM), whileCOM accelerates toward a cathode with its z -speed be-ing v COM ( t ) = − U t/z M , with t = 0 at the moment ofexplosion. Similarly to the Hubble expansion, the ion ra-dial distance ρ from COM is proportional to their originalvelocity, ρ = v t , v ≤ v max = p T max /M .By far, the most crucial “boost-factor” in the proposedscheme is the ion acceleration due to quasi- dc poten-tial. We consider the simplest neutron-producing reac-tion due to collision of two deuterium nuclei: d + d → He (0 . M eV ) + n (2 . M eV ). The importance of suf-ficiently high nuclei energy transpires from the fact thatthe fusion cross-section σ f ( ǫ in ), increases with ǫ in bymany orders of magnitude . Here ǫ in = U + T is afull energy of the collision of fast ion D + with D targetmolecules (such as e. g. D , CD , or D O [10,11]; D is the most efficient one [14]). Following [11] and usingfusion probability defined as w f ( ǫ in ) = Z ǫ in N tar vσ f ( ǫ ) dǫ/ ( dǫ/dt ) , (2)where dǫ/dt is the energy loss rate largely due to nucleus-nucleus scattering (the contribution of neutron genera-tion to this rate could be ignored), N tar is the numberdensity of target nuclei, and v is the velocity of deuterons.In the range of energies of interest and regular solid-statedensity, w f is approximated by a simple formula [11] w f ( ǫ in ) ≈ C D exp (cid:16) − p ǫ sc /ǫ in (cid:17) (3)where C D = 0 .
18 and ǫ sc = 7 M eV . Eq. (3) is best fittedfor D molecules; considering an example of cluster with N Σ ∼ , R = a (3 N Σ / π ) / with a ∼ − cm being an initial averaged spacing between ions (assumingan ideal case whereby all the atoms ionized by laser), wehave T max ∼ KeV , Eq. (1). In such a case, if onechooses U ∼ KeV , w f is boosted by ∼ . × .This makes it clear how far a Coulomb explosion is fromproducing a substantial neutron output. However, as im-pressive as this number may look, the probability w f perse is not the bottom-line to strive for, as far as energy nu-clear fusion is concerned. For that, the minimum goal isto have an averaged output energy of neutrons, ǫ D w f , toexceed that used to ionize deuterium and accelerate D + ions, ǫ in , so that the “in/out” efficiency η exceeds unity, η = ǫ D w f /ǫ in >
1, where ǫ D = 2 . M eV is the energy ofa neutron exiting fusion reaction. For real applicationsthe input power “at wall-plug” must be higher, but η > η vs in-put energy ǫ in is depicted in Fig. 2; one can see that η never reaches unity; it peaks at ǫ in = ǫ sc / . M eV [i. e. actually outside the domain of validity of Eq. (3)],and even then η max = 4( ǫ D C D /ǫ sc ) e − ≈ . ≪ . For ǫ sc < KeV , we have η < . ǫ sc < KeVη < − . Thus, the efficient fusion is unreachable here;essentially, this is true for any beam-target fusion, in-cluding laser-accelerated ion beams [15]. FIG. 2: The “in/out” efficiency of neutron production forbulk, η , and stack (multi-layered), η st , targets vs incidentenergy of deuteron nucleus. Inset: a schematic configurationof the stack target; ǫ st is the total potential across the stack. Barring the target compression (as in inertial con-finement fusion, with a required number density up to N ∼ cm − [10]), one way to enhance the target per-formance is to have it as a stack of thin multi-layers tobe crossed by ion beam, and apply a moderate dc voltagebetween them to replenish the energy lost by ions. Thisproposed “cascading” or “recycling” is to mitigate therapid ion energy loss in a bulk target. This loss resultsin dramatic reduction of fusion cross-section σ f ( ǫ ) in Eq.(3), so that a good part of the bulk’s volume is lost forthe neutron production. Making the layers thin enoughto have a low energy loss ∆ ǫ ≪ ǫ in per layer (to be re-plenished by an equal inter-layer potential), and theirnumber respectively large, N st = ǫ st / ∆ ǫ ≫
1, where ǫ st is the total potential across the stack, we have the prob-ability of fusion per layer as ( dw f /dǫ in )∆ ǫ , and the totalcascade probability w st ≈ ǫ st ( dw f /dǫ in ), such that η st = ǫ D w st ǫ st + ǫ in = η ( ǫ in ) r ǫ sc ǫ in ǫ st / ǫ st + ǫ in (4)If we choose e. g. ǫ st = ǫ in , the cascade target in therealistic setting offers the enhancement of p ǫ sc /ǫ in / η st ( ǫ in ) peaks at ǫ in = ǫ sc / ≈ KeV , with ( η st ) max = 0 . η st = 0 . ǫ in = 100 KeV , and 0 . ǫ in = 200 KeV . In spiteof enhancement, we still have η st <
1, i. e. the efficientfusion is still unreachable. Similar calculations for a D + ions and T ritium target, (with even higher output neu-tron energy, ∼ . M eV , instead of 2 . M ev for D + + D )leave this conclusion stand. As was mentioned, the sameis true for any beam+target fusion mode, as opposed to the inertial confinement fusion, whereby the efficiency η can in principle exceed unity. The major reason for this isthat the product of density and confinement time in theformer case is far below the one required by the so calledLawson criterion [10,11]; there is simply not enough timefor an D + ion to enter into a n -producing reaction in abulk target [16], while in a cascade target it needs energyreplenishment; the situation here is remotely reminiscentof a critical mass phenomenon in fission reactions. A po-tentially viable path for energy solution could be a hybridapproach combining ion-beam and inertial confinementfusion, but it is outside the scope of this Letter. A pos-sible way to go might be bombarding the solid-state tar-get by small ionized clusters, instead of separate ions, orchunks of partly-decomposed clusters sufficiently accel-erated by a E-pulse, which may create longer-sustainedhigh-temperature at the impact area (see also below),which we are planning to address elsewhere.Regardless of energy-producing prospects, the pro-posed system may be used both as a portable source ofneutrons, with the advantage of having orders of mag-nitude better time-clocking of the output neutron pulseover electrostatic sources [12] due to well synchronizedlaser pulse trigger, and a greatly useful research toolfor diagnostics of cluster structure and explosion cloud.While the diagnostics of ultra-short pulse laser-produced“macro”-plasma have been extensively studied (see e.g.[17]), the nanoscale mapping of f s pulse absorption wasstudied [18] recently using a ”plasma explosion imag-ing”. Yet combined with E -pulse, the system may def-initely offer new possibilities. Its diagnostics capabilitycomes naturally from the fact that it has already the ma-jor components (CE + E-field) employed by a so calledCoulomb Explosion Imaging [19-22] (CEI), used first in1989 [19] to yield images of small individual molecules.The CEI technique has been developed into a fine andsophisticated tool [20], including most recent use of pixel-imaging mass-spectrometer camera [21], and attainingthe first image of the Efimov trimer in helium [22]. How-ever, there is an important difference with cluster CE: acluster is comprised of too many atoms/ions, and asideperhaps from studying the details of its surface, whichmight be of interest to the physics of cluster formation,one is dealing with much more “macro-effects” than thosein the small individual molecules imaging, and might beless concerned about imaging. As an example, an impor-tant subject to explore is the cluster composition, to beelicited from analyzing the time dynamics of total ioniccurrent at the cathode, Fig. 1, as well as ”differential”currents flowing through sub-cathodes, see below. Theadvantage also is that usually a single laser shot and sin-gle cluster as e. g. in [8] and [18] can be used instead ofaveraging over many shots as in CEI.To have an idea of expected characteristics for thoseapplications, we consider some details of ion flow dy-namics. We introduce a potential/kinetic energy ratio, χ = U /T max , and a dimensionless time, τ = t/t ,where t = z p M/U is a time for COM to reach acathode due to potential U alone. The first ions hit thecathode in time τ min and the last ones – in τ max , where τ min,max = p χ − ∓ ∆ τ /
2; ∆ τ = 2 / √ χ and ∆ τ istotal duration of ion flow. Ions with the same startingenergy T make an expanding sphere (for its outer edgesee inset, Fig. 1); which falls down to a cathode withacceleration − U /z M . The rate number of ions hittingthe cathode at τ min < τ < τ max is as:1 N Σ dNdτ = 3 √ χ (cid:18) τ (cid:19) F ( τ ); F = Z ξ ξ f ( ξ ) dξ, (5)where ξ = T /T max is a relative initial kinetic energies ofions, and function f ( ξ ) describes a radial distribution ofthese energies that satisfies a condition R f ( ξ ) d ( ξ / ) =1. In the case of ideal CE, we have f CE = const = 1.To illustrate density/energy-profile sensitivity of this sys-tem, we will also consider two other distinctly differ-ent models of that distribution, in particular “hot ball”, f HB = 5 ξ/
3, which has a hollow core, while its outer shellis populated by hottest ions, thus making it a sustainedshock, and a “cool ball”, f CB = 5(1 − ξ ) /
2, with a densecold core and a vanishing density of “hot” outer ions.In Eq. (5), the integration limits ξ ( τ ) and ξ ( τ ) aredetermined by the area of the cathode engaged. If all theions hitting cathode are included, we have ξ = 1, and ξ = ξ min ( τ ) = ( χ/ (cid:0) τ − − τ (cid:1) <
1, which is a minimalinitial energy of ions reaching a cathode at the moment τ ,so that for CE, F CE = 1 − ξ min , for hot ball, F HB = 5(1 − ξ min ) /
6, and for a cool ball, F CB = 5(1 − ξ min ) /
4. Forthe CE case, the rate dN /dτ in units N Σ for χ from 2 . T max = 4 KeV , that wouldcorrespond to U ranging from 10 to 100 KeV ). Whileinitial kinetic energy of an ion is T , which increases to T = U + T when it hits a cathode, the total energy ofthe cloud delivered to the cathode during entire processin an ideal CE case, is T Σ = N Σ ( U + 3 T max / z up , between clusters and anode,(with total cathode-anode spacing z CA = z + z up ≥ z cr ,and thus a sufficient voltage between the plates, V CA ≥ V cr ), so that z cr /z = V cr /V = 1 + χ − . The max-imum “hot spot” radius ρ sp at the cathode is reachedat τ sp , which are respectively as: ρ sp /z = 2 √ χ/χ , τ sp = p χ − + 1, i. e. the spot gets tighter as χ in-creases, as expected. The effect is sensitive to the ionenergies distribution f ( ξ ) in the explosion, and thus mayoffer well-resolved time-of-flight diagnostics. It can beimplemented by segmenting the cathode into isolatedconcentric rings or sub-cathodes, and recording ion flowin each of them, as well as a total count of the ions, forvarious distributions f ( ξ ), i as illustrated in Fig. 3 forthree rings, The calculations here are based on Eq. (5),where we consider three model radial density profiles: (a) an ideal Coulomb explosion, f CE = const = 1, (b)“hot ball” profile, f HB = 5 ξ/
3, with a hollow core, whileits outer shell is populated by hottest ions, thus mak-ing it a sustained shock, and (c) a “cool ball” profile, f CB = 5(1 − ξ ) /
2, with a dense cold core and a vanish-ing density of “hot” outer ions. For the demonstrationpurposes we consider here the set of three rings, the cen-tral one being a disc with a radius ρ < ρ , a middle ring ρ < ρ c < ρ , and external ring ρ < ρ c < ρ max , wherewe set the sizes in such a way that in the end of the pro-cess, the total ion flow in each of them be the same for allthree sub-cathodes for the ideal CE case, whereby theyhave to be ρ /ρ sp ≈ .
475 and ρ /ρ sp ≈ . f ( ξ ) inthe original explosion, and thus may offer a well-resolvedtime-of-flight diagnostics of that distribution. Depend-ing on application, the electrodes geometry can be madeas spherical or cylindrical surfaces. In general case, in-stead of rings, the cathod can be made of multiple pix-els that would provide for much greater temporal andspatial resolution of explosion ion cloud and its dynam-ics. Ultimately, the cathode can be made as a large setof tiny pixels (with resolution determined then by pixelnumbers), as well as to serve as a mass-spectrometer toolin search and production of multi-atomic ”nano-chunks”to be explored in the attempt to bridge ion-beam andinertial confinement fusion.In conclusion, we proposed to reach ultimately highyield of neutrons in laser-driven explosion of deuteriumclusters, by using a synchronized E -pulse with up to100 KeV peak field, and making D + ions bombard a neg-atively charged deuterium-rich cathode. The maximumyield is reached as free electrons are removed from theion cloud, D + ions are fully utilized and made to collidewith solid-state deuterium-rich target instead of plasma.We also proposed to further enhance the output by usinga multi-layer target. While the energy production goalappears to be still unreachable, the major applicationof the system could be an efficient neutron source withlaser-controlled timing of neutron pulses. The system canalso be made into a sensitive diagnostic tool to resolve anintrinsic structure of ion cloud and cluster itself. ∗ Electronic address: [email protected][1] F. Floux, D. Cognard, L.-G. Denoeud, G. Piar, D.Parisot, J. L. Bobin, F. Delobeau, and C. Fauquignon,Phys. Rev. A , 821 (1970).[2] T. Ditmire, J. Zweiback, V. P. Yanovsky, T. E. Cowan, G. R a t e s o f i on f l o w ( d N / d t ) / N S i n s epa r a t e r i ng s Dimensionless time, t a Coulomb explosion N / N S N / N S t b ’Hot ball’ N / N S N / N S t c ’Cool ball’ N / N S N / N S FIG. 3: Rates of the flow of ions through sub-cathode rings (1 – inner, 2 – medium, 3 – outer rings), and time-integrated ioncount (inserts) in each of those rings for various models: (a) an ideal Coulomb explosion (CE) f CE = const = 1, with almosteven distribution of ions inside the cloud; (b) a “hot ball”, f HB = 5 ξ/
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