26-Sep-16. Nuclear energy production. Nuclear energy production. Nuclear energy production. Nuclear energy production

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1 Aim: valuat nrgy-gnration rat pr unit mass. Sun: (chck L /M, human ) nrgy-gnration rat producd from fusion of two nucli a + A: nrgy rlasd pr raction raction rat pr unit volum (includs cross sction and vlocity distribution ) Enrgy by ach raction - Most stars liv from so-calld thrmonuclar fusion, whr du to thrmal motion lightr nucli fus to form havir lmnts. - During fusion procss, som of th mass (.g. 0.7% for hydrogn burning) of original nucli has bn convrtd into nrgy according to. - Mass loss origins in diffrnt binding nrgis of th involvd nucli. - is nrgy rquird to sparat nuclons (protons & nutrons) against thir mutual attraction of th strong, short-rang forcs, or th gain if nuclons ar brought togthr (within cm) from infinity. total rst mass of nutrons total rst mass of protons mass of nuclus Intgr atomic wight: - Nuclar ractions in sun tak plac within innr ~10% of total solar mass. - Estimation of nrgy gnration by hydrogn burning ractions - mass loss: - nuclar tim scal: Enrgy by ach raction - for comparing nucli, bttr pr nuclon: Intgr atomic wight: -for A<56, f, bcaus of short-rang, strongforcs affcting only nuclons in its immdiat nighbourhood only (gomtrical ffct: surfac incrass with r slowr than volum, i.. with A). -for A>56, f, bcaus of rpulsiv Coulomb forcs btwn protons, which ar far-raching F most tightly bound nucli (smallst m/nucli). - any raction bringing rsulting nucli clos to f max will b xothrmic: (a) for A<56 by fusion (.g. stllar cors) (b) for A<56 by fission (.g. radioactivity) 1

2 Enrgy by ach raction - Considr gnral raction: Short notation: - Th nrgy rlasd in this raction is: - Sinc loss of mass is gnrally vry small, mor convnint to us mass xcss protons nutrons atomic mass xcsss availabl in form of tabls (.g. Clayton 1968). Th cross sctions Masurd cross sction for raction Stllar cor conditions ~25 kv Enrgy (MV) Th cross sctions dfind as an ara, which dscribs th rlation btwn probability p of a raction and a gomtrical ara. - - raction btwn nucli is causd by th strong forcs, acting btwn protons and nutrons (rang limitd to xtnt of nuclus). - for a raction to occur, nucli must b brought so clos togthr that thy touch ach othr rquirs to ovrcom Coulomb rpulsion btwn thm. - nrgy to ovrcom Coulomb rpulsion,i.. Coulomb forc hight of Coulomb barrir E CB r~10-8 cm - howvr, avrag kintic nrgy of particls intraction radius r 0 =r nucl ~A 1/ cm nuclar (strong) forc Th cross sctions - rasonabl xtrapolation of to lowr can b obtaind by sparating strongst -dpndnt contributions causd by (quantum-mchanical) pntration probability through Coulomb barrir. - quantum-mchanical ffct found by G. Gamov (1928) is th tunnl-ffct, in which nucli can pntrat (tunnl) th potntial barrir with a probability rlativ vlocity btwn th two nucli: v ~ E 1/2 - an additional - dpndnc ntrs in th gomtrical xtnt of th nucli: d Brogli wavlngth gomtrical cross sction 2,

3 Th cross sctions - cross-sction factor (astrophysical cross sction) astrophysical gomtrical factor Coulomb barrir - hr is nrgy of particls in th cntr-of-mass systm rducd atomic wight whr Ractions amongst nucli of low chargs ar fastr and ar possibl at low tmpraturs. astrophysical cross sction -dpartur from (nar) constant valu at low E rsults from scrning from lctrons in th targt 6 Li or 7 Li atoms, which rducs Coulomb potntial around nuclus, and hnc incrass Th cross sctions cross sction factor - w nd rlation btwn and total rat of ractions, btwn particls a and A Probability that on particl a, th projctil, will hav a collision along dl is i.. - particl flux of projctil: - ractions pr unit tim in whol box: projctils passing da vry scond - ractions pr unit tim and volum: - spcial cas A=a : 3.

4 - In rality A is not at rst and thr is a distribution of th rlativ vlocitis v. If f(v)dv is th fraction of pairs of particls with rlativ spd btwn v & v+dv with - W assum that both nucli hav M-B distribution, thn (s Clayton 1968) th distribution of v is also M-B whr rducd mass with rducd atomic wight. - analytical approximation of Gamov pak: -w nglct variation of S(E) ovr Gamov pak [S(E) S(E 0 )], and approximat th function by a Gaussian whr th width provids th sam curvatur at maximum. Aftr som algbra w obtain with For fairly larg, is a dcrasing function of : for. and w obtain intrinsic nuclar ractions includs gomtry of particls M-B factor of thrmal nrgy Coulomb barrir (tunnling) compting ffcts rsulting in Gamov pak maximum at - in ordr to s th T-dpndnc mor clarly, w furthr simplify th xprssion ovr a limitd T rang around T=T 0, dfining T 0 such that and adopt th powr law Bcaus, th T-snsitivity of incrass strongly with :.g. n= for H-burning in Sun n=13 23 for CNO-cycl. 4.,,

5 W may now writ th raction rats (ractions /s /cm 3 ) as Th nrgy gnration rat pr unit mass is thn whr w usd Rat of chang in du to ractions with nucli of typ abundancs by mass - particl dnsity n with charg q is modifid in th prsnc of an lctrostatic potntial according to # dnsity for = 0 drivd from Dby-Hückl thory (s.g. Jackson 1975) - Typically approximation: Which shows dirctly th dcras of ion dnsity and incras of - dnsity. - rpulsiv Coulomb forc is important for stimating. - fr - hav influnc on Coulomb forc, i.. E CB. - approaching particl will fl nutral conglomrat of targt nuclus & surrounding - - cloud. - - ar attractd from nuclus of charg +Z - hav slightly n nar nuclus, i and ions ar rplld, i.. n clustring of -. - total (lctrical) charg dnsity for all typs of ions (n i ) (nutral gas) i..: - Combin last two trms to obtain whr ar 5., :

6 - and ar also connctd via th Poisson quation which has th solution (for a point charg Z) and for sphrical gomtry with bing th Dby-radius (to som sns th r of - cloud; r D = cm.) For normal stars classical particl distant at Gamov pak unshildd potntial - wak scrning: For solar cas: - strong scrning: Scrning factor dpnds strongly on & T - Accordingly th Coulomb barrir E CB is rducd by Dby nrgy E D Dby nrgy E D Thrby incrasing th probability with which particls can tunnl through th Coulomb barrir, lading to an incras of Coulomb barrir M-B tail of thrmal nrgy scrning factor f Incras by f : f maintain charg balanc (consrv lpton numbr) (a) pp chain: no 4-body ractions sub-ractions (controlld by wak intraction) 2x T ~ 15 mill. K 6..

7 maintain charg balanc (consrv lpton numbr) (a) pp chain: no 4-body ractions sub-ractions (controlld by wak intraction) 2x T ~ 15 mill. K (a) pp chain: no 4-body ractions sub-ractions shilding factor maintain charg balanc (consrv lpton numbr) (a) pp chain: no 4-body ractions sub-ractions (controlld by wak intraction) 2x T ~ 15 mill. K (b) CNO cycl: (C, N & O ar catalysts) positron dcay (p n) CNO (I) CNO(II) [10 4 tims lss likly] T ~ 20 mill. K 7

8 (b) CNO cycl: (C, N & O ar catalysts) positron dcay (p n) CNO (I) CNO(II) [10 3 tims lss likly] T ~ 20 mill. K Hlium burning Tripl raction: prformd in 2 stps: (dcays aftr s) Onc sufficint is producd, additional to tripl raction, succssiv -caption: Also: (b) CNO cycl: Carbon burning Whn H is xhaustd, nxt lmnt to ract is with Scrning factor may bcom important. (ndothrmic raction) most probabl 8

9 Oxygn burning Coulomb barrir alrady so high that tmpraturs ar rquird. most frqunt 2 nd most frqunt Photodisintgration and subsqunt captur: (T so high that photons in MV rang; ths may b absorbd by nucli, causing thm to brak up, typically in an dcay),.g. non disintgration (burning) particl usd in H burning: (ndothrmic occurs bfor oxygn burning) nt raction: Silicon burning: two 28 Si 56 F nd suprnova xplosion. Nutrinos Hav xtrmly small cross sction Nutrinos in MV rang hav man fr path ( =1) Normal stllar mattr : : collapsing stllar cor: rabsorbd within star. Also important for cooling cors of stars. Nutrinos (nrgy loss du to scaping nutrinos): With ~ 4x10-5 rg/cycl Sun producs ~ 2x10 38 (lctron) nutrinos vry scond! Flux of ~ solar nutrinos pr cm 2 and scond at Earth! Urca procss at xtrm dnsitis (no nuclar ractions ncssary): - captur: Nuclus with charg Z and atomic wight A dgnrat 9

10 Nutrinos othr procsss without nuclar ractions onc in tims annihilation : E- EM wav: disprsion q.: coupls with - of - 10