Enhanced electron screening in nuclear reactions and radioactive decays

Transcription

1 nhancd lctron scrning in nuclar ractions and radioactiv dcays 1, *, P. Hid, A. Huk, L. Martin,. Ruprcht 3 1 Institut of Physics, Univrsity of Szczcin ul. Wilkopolska 15, Szczcin, Poland Institut für Optik und Atomar Physik, Tchnisch Univrsität Brlin Hardnbrgstr. 36, 1063 Brlin, rmany 3 TRIUMF National Laboratory 4004 Wsbroock Mall, Vancouvr, BC, V6T A3, Canada -mail: czrski@kalium.physik.tu-brlin.d In rcnt yars, an nhancd lctron scrning in mtallic nvironmnts has bn dmonstratd by many groups in xprimntal invstigations of low-nrgy nuclar ractions. Similarly, first radioactiv dcay xprimnts in mtallic matrials hav bn prformd to possibly obsrv an altration of th dcay constant du to lctron scrning. Both kinds of xprimnts ar of fundamntal importanc for nuclar astrophysics sinc th mtallic quasi-fr lctrons rprsnt a modl for dns astrophysical plasmas and thus th corrsponding thoris can b xprimntally vrifid. Hr, th xprimntal rsults dtrmind for d+d fusion ractions will b applid to propos a scaling procdur for lctron scrning nrgis in diffrnt mtallic matrials. Th rsults will b compard with th xprimntal valus obtaind for ractions with havir nucli and som prdictions for radioactiv dcay xprimnts will b prsntd. Additionally, th tmpratur dpndnc of th scrning nrgy will b discussd. Intrnational Symposium on Nuclar Astrophysics Nucli in th Cosmos IX CRN, nva, Switzrland 5-30 Jun, 006 * Spakr Copyright ownd by th author(s) undr th trms of th Crativ Commons Attribution-NonCommrcial-SharAlik Licnc.

2 nhancd lctron scrning 1. Introduction Alrady first obsrvations of th nhancd lctron scrning ffct in th d+d fusion ractions in mtallic nvironmnts hav bn motivatd by its importanc for dns astrophysical plasmas [1]. From th thortical point of viw th dutrizd mtals can b tratd as a strongly coupld plasma and thrfor provid a uniqu possibility for xprimntal studis of nuclar pycnoractions undr trrstrial conditions. Th thortical calculations [] basd on th slf-consistnt dilctric function thory can qualitativly xplain th targt matrial dpndnc of th xprimntally dtrmind scrning nrgis, howvr, th absolut thortical valus ar undrstimatd by a factor of two. Th rason for this discrpancy is still unknown. Larg scrning nrgis in mtallic nvironmnts hav bn also dmonstratd by othr groups [3,4]. Som diffrncs btwn xprimntal rsults obtaind by diffrnt groups probably rsult from a targt surfac contamination or an inhomognous distribution of dutrons within th targt [5]. nrally, th xprimntal scrning nrgis rach about 300 V for havir mtals and ar much smallr (blow 100 V) for insulating matrials. Th rsults obtaind for d+d ractions can b vrifid in ractions with havir nucli. Rcntly, larg scrning nrgis hav bn obsrvd for th 6 Li+d systm [6] and confirmd by th LUNA collaboration [7]. In th prsnt work, w would lik to provid a scaling procdur allowing to prdict th scrning nrgis for havir raction systms. On th othr hand, th sam procdur can b applid for studis of radioactiv dcays taking plac in diffrnt nvironmnts. Larg diffrncs btwn th scrning nrgis for insulating and mtallic nvironmnts nabl to anticipat a corrsponding half-lif altration for alpha and bta dcays. Additionally, a possibl tmpratur dpndnc of th scrning nrgy as proposd in [8, 9] will b discussd.. lctron scrning in nuclar ractions with havy nucli Th lctrostatic potntial nrgy btwn racting nucli of chargs Z 1 and Z in a mtallic mdium can b dscribd in analogy to dns astrophysical plasmas within th slfconsistnt dilctric function thory [5]: Z Z Z Z 4π ϕ1( q) ϕ( q) 1 1 rr 3 Z1Z V ( r) = Φ( r) = xp( iqr) d q U 3 r (π ) ε ( q) ε ( q) q r Th wav-numbr dpndnt dilctric functions v c ε v and (1) ε c dscrib polarization of valnc and cor lctrons of host atoms inducd by a chargd impurity. Φ(r) and ϕ i (q) functions ar th scrning function and lctronic charg-formfactors of racting nucli, rspctivly (for dtails s []). At small distancs (applicabl for nuclar ractions and dcays) th potntial nrgy can b approximatd using th nrgy indpndnt scrning nrgy U. From q.1 it can b found that th scrning nrgy scals with th product of th chargs of th involvd

3 nhancd lctron scrning nucli. Th rsults obtaind for th d+d ractions taking plac in diffrnt mtallic targts indicat that th lctron scrning nrgis modratly incras with th atomic numbr of host atoms [5] raching for havir nucli th valu of about 300V. This targt matrial dpndnc agrs with th thortical xpctations and mainly rsults from an incrasing cohsion nrgy diffrnc btwn two racting dutrons and th compound nuclus 4 H mbddd in th crystal lattic of havir mtals (so-calld cohsion scrning). On th othr hand, th xprimntal scrning nrgis ar largr by a factor of about compard to th thortical valus. Sinc th xprimntal scrning nrgis obtaind for insulating matrials ar much smallr (<100V) [4,5] and taking into account that th scrning contributions rsulting from polarization of bound host lctrons and cohsion should b similar for both mtallic and insulating targts, w can conclud that th nhancd scrning ffct ariss from conducting lctrons. In th Thomas-Frmi modl this contribution is givn by U,f (TF) =Z 1 Z [4/(πa 0 )] 1/ (3π n) 1/6 whr n and a 0 ar th valnc lctron dnsity and th Bohr radius, rspctivly. Using th lctron-gas paramtr r s =[3/(4πn)] 1/3 /a 0 w obtain U,f (TF) =Z 1 Z [9/(4π )] 1/6 r s -1/ = Z 1 Z 4.5V / r s 1/. Th sam scaling can b applid for th xprimntal valus dtrmind for Li, Al, Zr, Pd and Ta targts [5]. In ordr to stimat th xprimntal contribution of fr lctrons th xprimntal scrning nrgis hav to b rducd by contributions arising from bound lctrons and th cohsion scrning. Finally, w gt U,f = Z 1 Z (50±0) V / r s 1/, which can b usd for an stimation of th fr lctron contribution in mtallic nvironmnt to th scrning nrgy in ractions btwn nucli with chargs Z 1 and Z. Diffrnt to th d+d ractions, th contribution coming from lctrons bound by havir racting nucli is much largr and should b includd sparatly. This can b calculatd as th gain in lctron binding nrgis btwn distant atoms and th final unitd atom. Similar rsults ar to obtain using th Thomas-Frmi modl lading to U,b (TF) = 1.13 Z 1 Z (Z 1 1/ + Z 1/ ) /3 /a 0 [10]. Th total scrning nrgy is th sum of th fr lctron and bound lctron contributions. Th first xprimntal dtrmination of th scrning nrgy in mtallic nvironmnts for havir raction systms ( 6 Li+d) was prformd by Kasagi t al. [6] and confirmd latr by th LUNA collaboration [7]. In Tab.1 th xprimntal rsults for th 6 Li+d, 6 Li+p and 7 Li+p systms in th insulating (LiF) and mtallic (Li and PdLi x alloy) nvironmnts ar prsntd and compard to th thortical valus valuatd according to th scaling givn abov. Th scrning nrgy for bound lctrons calculatd as th gain in th lctron binding nrgis amounts to 190 V and agrs with th Thomas-Frmi stimation of 180 V as wll as with th Tab.1 Scrning nrgis for 6 Li+d, 6 Li +p and 7 Li+p systms host scrning nrgy (V) matrial bound fr tho (bound+fr) xp insulator ,11 Lithium PdLi x

4 nhancd lctron scrning. nhancmnt Factor α =5 MV r s = scald dd scrning Thomas-Frmi scrning insulator Atomic Numbr Fig. 1 nhancmnt factors for alpha-dcay ( α =5 MV) in mtallic (r s =) and insulating nvironmnts xprimntal rsults. Th fr lctron contribution to th scrning nrgy is diffrnt for mtallic Li and Pd nvironmnts bcaus of diffrnt lctron dnsitis. A comparison btwn thortical and xprimntal scrning nrgis is prsntd in Tab.1. Th rrors of th xprimntal scrning nrgis ar rlativly larg mainly du to systmatic uncrtaintis. Nvrthlss, both thortical and xprimntal data confirm a strong targt matrial dpndnc and much largr scrning nrgis than for th d+d ractions. 3. lctron scrning in radioactiv dcays In radioactiv dcays, whr th mittd particl intracts with th rcoil nuclus lctrostatically, th lctron scrning ffct plays an important rol. In alpha dcays alpha particls hav to tunnl th Coulomb barrir in opposit dirction compard to low nrgy nuclar ractions. Thus, th probability for alpha-dcay dpnds on th lctronic structur of both th dcaying atom and th host mdium. Assuming that th chang of th Coulomb barrir du to surrounding lctrons can b dscribd by a scrning nrgy U and th binding nrgy of th dcaying nuclus rmains unaffctd w obtain an xprssion for an incras of th dcay probability compard to th bar nuclus cas as follows: f = P( + U ) P( ) = + U xp xp + U U xp () 4

5 nhancd lctron scrning.8.4 Z=8 r s = nhancmwnt Factor scald dd scrning Thomas-Frmis scrning insulator α (MV) Fig. nhancmnt factors for alpha dcay (Z=8 MV) in mtallic (r s =) and insulating nvironmnts Hr, th nhancmnt factor f is th ratio of th corrsponding pntration factors for scrnd and bar nucli. Th approximation in q. () can b mad whn th alpha dcay nrgy is much largr than th scrning nrgy U. Th amow nrgy is givn by 4πZ = hc whr µ dnots th rducd mass. In th cas of mtallic nvironmnts, th scrning nrgy U consists of two componnts: on rsults from bound lctrons of th dcaying atom and th othr from quasi-fr conduction lctrons of th mtal. Similar to nuclar ractions with havy nucli th cohsion scrning contribution as wll as polarization of bound lctrons of th host atoms can b nglctd. For insulating nvironmnts w only hav th bound lctron contribution. Sinc th binding nrgis of innr atomic lctrons dpnd only vry slightly on th crystal lattic structur, w assum that this contribution has th sam form for insulating and mtallic matrials and can b dscribd by th Thomas-Frmi modl. Th nhancmnt factor for th alpha dcay with α = 5 MV calculatd for mtallic (r s =) and insulating nvironmnts is prsntd in Fig.1: Th highr atomic numbr of th dcaying nuclus th largr diffrnc btwn half-livs in insulators and mtals. For 38 U and α = 5 MV w gt f=1.5 for insulating and f=1.31 or 1.61 for mtallic (r s =) nvironmnts using Thomas-Frmi modl or d+d scaling. In Fig. th dpndnc of th nhancmnt factor on th alpha-dcay nrgy is prsntd. Th nrgy spctrum of mittd positrons (+) and lctrons (-) in allowd nuclar btadcays can b writtn as follows: µ 5

6 nhancd lctron scrning dn ± = C M F ± ( Z, W ) pw ( W W ) 0 dw (3) whr C, M and F ± ar a constant factor, th transition matrix and th Frmi function for positrons or lctrons, rspctivly. W and p dnot th total nrgy and th momntum of bta particls, W 0 corrsponds to th maximum nrgy. Th Frmi function dscribs Coulomb intraction btwn th bta particls and th dcaying nuclus and changs th nrgy spctrum as wll as th total dcay probability such that ß - dcay is mor probabl than ß +. Whn th Coulomb barrir is shildd by surrounding lctrons th total ß - dcay probability dcrass (half-lif incrass) and th ß + dcay probability incrass (half-lif dcrass). Numrically, th lctron scrning can b takn into account by rplacing th total lctron nrgy W in th Frmi function into W =W+U as follows [1]: p W F ± scr ( Z, W ) = F ± ( Z, W ) (4) pw Th scrning nrgy U can b calculatd similar to alpha dcay. Th influnc of th scrning ffct on bta dcay rats is not so strong as for alpha dcay. In Fig.3 th nhancmnt factor (ratio btwn th dcay probability for scrnd and bar nuclus) for ß + dcay of W 0 =1.511 MV ( β =1 MV) is prsntd. Although th nhancmnt factors ar rlativly small thy should b dtrminabl xprimntally β =1 MV r s = nhancmnt Factor scald dd scrning Thomas-Frmi scrning insulator Atomic Numbr Fig. 3 nhancmnt factors for ß + nvironmnts dcay ( ß =1 MV) in mtallic (r s =) and insulating 6

7 nhancd lctron scrning 4. Tmpratur dpndnc of th scrning Th suggstion [9] that th nhancd lctron scrning in mtallic nvironmnts could b dscribd by th Dby-Hückl thory would imply a strong tmpratur dpndnc of th scrning nrgy 1 / T. As alrady pointd out in [5] th Dby-Hückl scrning (so-calld wak scrning) is applicabl only for high tmpraturs or quivalntly for high projctil nrgis. At lowr tmpraturs corrsponding in mtallic nvironmnts to vlocitis blow th Frmi vlocity th lctron scrning rachs th strong-scrning limit. Th scrning nrgy dpndnc on th projctil nrgy (or on tmpratur according to th quation µv /=3/kT) calculatd for th d+d fusion ractions in Al is prsntd in Fig.4. Th transition from th wak scrning to th strong scrning taks plac at th Frmi nrgy of 0 kv for Al and corrspondingly at th Frmi tmpratur of about 10 5 K. For lowr nrgis or tmpraturs th scrning nrgy changs only vry slightly (blu curv). Scrning nrgy (arb. units) 1 Tmpratur (K) Al T F strong scrning wak (Dby) scrning F Dutron nrgy in CMS (kv) Fig.4 Scrning nrgy dpndnc on th tmpratur rsp. projctil nrgy 5. Conclusions and discussion Larg xprimntal valus of scrning nrgis obtaind for d+d fusion ractions provid intrsting consquncs for nuclar ractions with havir nucli whr th scrning nhancmnt ffct should b strongr. Basd on dilctric function thory som thortical stimation can b mad for which a scaling of th scrning nrgy with th atomic numbr has bn proposd. Diffrncs in th scrning nrgis for insulating and mtallic nvironmnts lad to a prdiction of altration of half-livs for radioactiv dcays in various mdia. spcially for alpha dcays th shortning of half-lif du to th lctron scrning should b asy to prov. Howvr, an additional nrgy shift of th alpha dcaying stat du to th lctron 7

8 nhancd lctron scrning scrning, which has not bn includd in th prsnt analysis, can rduc th prdictd nhancmnt. Th scrning ffct in bta dcays, although wakr, could also b xprimntally vrifid. Thus, futur xprimntal studis of th scrning ffct in radioactiv dcays offr a nw possibility to clarify discrpancis btwn thortical and xprimntal scrning nrgis obsrvd in low-nrgy nuclar ractions. An indpndnt confirmation of th obsrvd nhancd scrning ffct in mtallic nvironmnts should hav far-raching consquncs for astrophysical plasmas as wll as for som tchnical applications, for instanc for th fusion-ractor tchniqu. Th Dby tmpratur dpndnc of th scrning nrgy proposd in [8,9] is not applicabl for mtals. Howvr, som small ffcts du to changs of lctronic band structur might b xpctd (s also th xprimntal invstigation of ß + dcay of Na [13]). Rfrncs [1], A. Huk, P. Hid, M. Hoft,. Ruprcht, Nucli in th Cosmos V, Procdings of th Intrnational Symposium on Nuclar Astrophysics, July 6-11, 1998 Volos, rc, ditd by N. Prantzos and S. Harissopulos (ditions Frontièrs) p.15, A. Huk, A. Billr, P. Hid, M. Hoft,. Ruprcht, urophys. Ltt. 54 (001) 449 [], A. Huk, P. Hid,. Ruprcht, urophys. Ltt. 68 (004) 363 [3] J. Kasagi t al., J. Phys. Soc. Jpn. 71 (00) 81 [4] F. Raiola t al., ur. Phys. J. A 13 (00) 377 [5], A. Huk, P. Hid,. Ruprcht, ur. Phys. J. A 7 (006) 83 [6] J. Kasagi t al., J. Phys. Soc. Jpn. 73 (004) 608 [7] J. Cruz t al., Phys. Ltt. B 64 (005) 181 [8] F. Raiola t al., J. Phys. 31 (005) 1141 [9] B. Limata t al., ur. Phys. J. A 8 (006) 51 [10] J.F. Ziglr, J.P. Birsack, U. Littmark, Th Stopping and Rangs of Ions in Solids, Prgammon Prss, Nw York, 1985 [11]. Ruprcht,, D. Bmmrr, M. Hoft, P. Hid, Phys. Rv. C 70 (004) [1] L. Durand, III, Phys. Rv. 135 (1964) B310 [13]. Ruprcht t al., this confrnc. 8