C(p, ) 13 N. Nuclear reactions generate energy create new isotopes and elements. Notation for stellar rates: p 12

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1 Iroducio o sellar reacio raes Nuclear reacios geerae eergy creae ew isoopes ad elemes Noaio for sellar raes: p C 3 N C(p,) 3 N The heavier arge ucleus (Lab: arge) he ligher icomig projecile (Lab: beam) he ligher ougoig paricle (Lab: residual of beam) (Lab: arge) (Lab: beam) (Lab: residual he heavier residual ucleus (Lab: residual of arge) (adaped from radiioal laboraory experimes wih a arge ad a beam)

2 Typical reacios i uclear asrophysics: (p,g) (p,) (p,) (,) (,) : (B,) (B,p) (B,) (B,) ad heir iverses

3 cross secio bombard arge uclei wih projeciles: relaive velociy v Defiiio of cross secio:. # of reacios = # of icomig projeciles per secod ad arge ucleus per secod ad cm or i symbols: = j wih j as paricle umber curre desiy. Of course j = v wih paricle umber desiy ) Uis for cross secio: bar = 0-4 cm ( = 00 fm or abou half he size (cross secioal area) of a uraium ucleus)

4 Reacio rae i sellar evirome Mix of (fully ioized) projeciles ad arge uclei a a emperaure T Reacio rae for relaive velociy v i volume V wih projecile umber desiy p R p p v v T V so for reacio rae per secod ad cm 3 : Reacios per secod r p T v This is proporioal o he umber of p-t pairs i he volume. If projecile ad arge are ideical, oe has o divide by o avoid double couig ( ) as here are pairs per volume, herefore r pt p T v

5 Relaive velociies i sars: Maxwell Bolzma disribuio for mos pracical applicaios (for example i sars) projecile ad arge uclei are always i hermal equilibrium ad follow a Maxwell-Bolzma velociy disribuio: he he probabiliy (v) o fid a paricle wih a velociy bewee v ad v+dv is ( v) 4 3/ m v m kt ( v) dv kt 4 v e wih arbi rary uis 3 Max velociy correspods To E=kT example: i erms of eergy axis E=/ m v eergy (kev)

6 oe ca show (Clayo Pg 94-95) ha he relaive velociies bewee wo paricles are disribued he same way: ( v) 4 kt 3/ v kt wih he mass m replaced by he reduced mass of he paricle sysem v e m m m m he sellar reacio rae has o be averaged over he disribuio (v) r or shor had: pt r p T ( v) ( v) vdv pt p T v ypical srog velociy depedece!

7 expressed i erms abudaces r pt T p N v reacios per s ad cm 3 pt p N v reacios per s ad arge ucleus his is usually referred o as he sellar reacio rae of a specific reacio uis of sellar reacio rae N <sv>: 3 usually cm /s/mole, hough i fac cm 3 /s/g would be beer (ad is eeded o verify dimesios of equaios) ( does o have a ui)

8 budace chages, lifeimes, eworks Les assume he oly reacio ha ivolves uclei ad B is desrucio (producio) of (B) by capurig he projecile a: + a -> B d les assume he reacio rae is cosa over ime. This is a very simple reacio ework: B Each isoope is a ode ha is liked o oher isoopes hrough producio ad desrucio chaels Sarig from a iiial abudace, we ca he ask, how he abudace of each ework ode evolves over ime Typically he same ligh projeciles drive mos of he reacios (euro or proo capure) so we do eer p, ad all is desrucio chaels io he graphics bu udersad ha hey ge produced ad desroyed as well)

9 We ca wrie dow a se of differeial equaios for each abudace chage d a N d db d v ssumig, he reacio rae is cosa i ime, his case ca be solved easily (same as decay law): ( ) B ( ) 0 0 e ( e )

10 ad of course Example: B ( ) ( ) 0 0 e ( e ) afer some ime, ucleus is eirely covered o ucleus B abudac ce B same abudace level /e ime Lifeime of (agais desrucio via he reacio +a) : (of course half-life of T / =l/ a v N

11 Eergy geeraio hrough a specific reacio: Reacio Q-value: Eergy geeraed (if >0) by a sigle reacio i geeral, for ay reacio (sequece) wih uclear masses m: Q c m i m j iiial uclei i fial uclei j Eergy geeraio Eergy geeraed per g ad secod by a reacio +a: rq Q a a N v Ui i CGS: erg ( erg = E-7 Joule) (remember, posiro emissio almos always leads o a addiioal eergy release by he subseque aihilaio process ( x.5 MeV))

12 Reacio flow abudace of uclei of species covered i ime i ime ierval [,] ] io species B via a specific reacio B is called reacio flow d FB d B() () d d B For Ne reacio flow subrac he flow via he iverse of ha specific reacio (his is wha is ofe ploed i he ework coecig he odes) F F F e B B B (Someimes he reacio flow is also called reacio flux) I our example, a ifiie ime has bee covered eirely io B. Therefore F e B ( ) ( 0)

13 Muliple reacios desroyig a uclide example: i he CNO cycle, 3 N ca eiher capure a proo or decay. 4 O (p,) each desrucive reacio i has a rae i 3 N ( + ) Toal lifeime he oal desrucio rae for he ucleus is he is oal lifeime i i i i 3 C Brachig he reacio flow brachig io reacio i, b i is he fracio of desrucive flow hrough reacio i. (or he fracio of uclei desroyed via reacio i) b i i j j

14 Geeral reacio ework se of isoopes wih abudaces i, Cosider - ad -body raes oly d d i j,k j k N v jki l li l producio m i m N v imay i i desrucio Noe ha his depeds o mass desiy ad emperaure (hrough <v> ad ) so his requires ipu from a sellar model. Needs o be solved umerically. This is o rivial as sysem is very siff (reacio rae imescales vary by may may orders of magiude)

15 Example for a more complex ework (rp-process process i X-ray burss) Mass kow < 0 kev Mass kow > 0 kev Oly half-life kow see Figure: Schaz&Rehm, Nucl. Phys.,