Nuclear Level Density Calculation of Astrophysical Reactions

Transcription

1 IOSR Joural of pplied Pysics (IOSR-JP) e-issn: Volume 8, Issue 4 Ver. I (Jul. - ug. 2016), PP Nuclear Leel Desity Calculatio of stropysical Reactios med. Selma 1, Hamsa S. Jasim 2 1 Departmet of stroomy ad Space, College of Sciece, Bagdad Uiersity, Bagdad, Iraq 2 Departmet of stroomy ad Space, College of Sciece, Bagdad Uiersity, Bagdad, Iraq bstract: Nuclear state desity was calculated for te uclear reactio iside te core of tree differet maisequece stars amely, Su, Sirius ad Vega. Te selected uclei for preset calculatio were H 4, C 12, N 14, ad O 16 ; ad te state desity of Fe 54 ucleus was also icluded for compariso. Te state desity was calculated based o te excito teory of uclear reactio from Ericso, Williams ad pairig formulae. Te results cofirmed tat ay cage i te cofiguratio will result differet alues of state desity ad te results sowed tat te depedece o eergy is ot liear, ad iger desities were foud at iger eergies, but te cage was less. I. Itroductio Te pysics of uclear structure ad uclear reactios is eterig a ew field, due to te deelopmet of radioactie uclear beams, wic eable us to study a ariety of uclei far from stability. uclear reactio caused by a icidet particle result i sarig of te icidet particles eergy wit all ucleos witi te ucleus to reac te fial stage is called "Termal Equilibrium State". Te ucleus at tis stage is called "Te Compoud Nucleus". I betwee direct ad CN emissios, tere is a itermediate ad cotiuous emissio of particles wic is called te PE.. Griffi [2] i 1966 proposed te excito model, wic is a semi-classical approac to explai PE, Beig a simplified model. importat parameter i PE teory is te state desity. Te importace of te state ad leel desities rise from its ability to describe populatio of uclear excited states. Te aim is to try applyig te excito model o stellar reactios. Studies of te solar structure ae sow tat te mai eergy it radiates comes from a series of uclear reactios called te p-p cai reactio ad CNO cycle [1-7]. II. Te Excito Model Griffi [2] assumed te precompoud ucleus will exist wit formed Excitos. te mai reasos beid te popularity of Griffi s excito model are [3]: 1. Excito creatio meas creatig te pair (particle +ole), te particle beig excited aboe F. 2. Eac excito state may decay wit a time rate tat depeds o te excito umber,, ad excitatio eergy, E. I uclear reactios, tis must be te case because te decay rate must deped o umber of excited particles (populatio of excited particle states) ad excitatio eergy. 3. We may umbers of excitos are created, te stregt of te excito creatio process becomes less sigificat. Tis meas tat as te excitatio states deeloped ad grow i umber, excito creatio becomes less effectie due to te reductio of te excitatio eergy wic lowers te caces of excito creatio. 4. Tere is a maximum excito umber max wic represets te fial possible excito state. I uclear reactios, also a fial excitatio state must exist. t max, o furter excito creatio is possible. I terms of uclear reactios, it is said tat te ucleus reaced termal equilibrium ad te CN is formed. 5. Tus, coosig excito idea to represet uclear reactios is quite suitable. III. Nuclear Reactios i Mai Sequece Stars Te solar structure as sow a series of uclear reactios called te p-p cai ad CNO cai reactios [10-12]. Te p-p reactio comes from a series of te followig idiidual reactios or steps: Te first ad tird reactios could take 10 6 years to start, wile te secod step occurs witi 10-3 sec. Eac step of tese produces a certai amout of eergy from mass to eergy coersio E=m o c 2, because te elium ucleus weigs less ta te sum of te four protos. Eergy emerges i tree forms: eutrios tat flee DOI: / Page

2 te Su's core at te speed of ligt, protos tat participate i a ew roud of fusio, ad potos tat percolate troug te Su (still at te speed of ligt) util tey leae te Su's surface as isible ligt. CNO is a fusio process tat combies protos to form elium uclei troug itermediary uclei of carbo, itroge, ad oxyge. Tis process requires a temperature of about 15 millio K, wic is iger ta te core temperature of te Su, but is less efficiet ta te Su's p-p cai reactio fusio reactio. Te reactios are as follows: Te process, terefore, icludes cosumig four ydroge atoms. Te carbo cycle requires ig temperature to operate, wic ca oly foud i te core of massie stars [1]. IV. Te State Desity Te state desity is a importat quatity were te mai calculatios are based o. Te Equidistat- Spacig Model (ESM) approximatio of te uclear state desity is described by te sigle particle state desity g. I ESM, te eergy differece betwee ay two successie eergy leels ise = 1/g = costat [4]. Te excito umber,, is defied as =p +, we it is assumed tat protos ad eutros are idistiguisable particles. For a two-compoet Fermi gas system, we ae = + p + + p + Te state desity for oe-compoet Fermi gas system will be deoted by (,E) ad for two-compoet by (,E). Te state desity calculatio icludes te followig treatmets: 1- U-corrected (Ericso) formula, 2- Pauli blockig (Williams Formula), 3- Back-sifted leels ad modified Pauli terms, 4- Pairig correctio. 4.1 Ericso s Formula Te simplest formula used to describe (,E) is to exclude all possible correctios durig te calculatios, ad it is called Ericso Formula see [3]. Tis formula sows tat state desity is gie for oecompoet system by, (, E) 1 1 g E p!!( 1)! Were g is te sigle particle leel desity [5] g= most practical calculatios. For two-compoet system, te state desity is gie as, (, E) 2 p! 1 g E! p!! ( 1)! DOI: / Page (1) ) or g to, wic is used i wic is writte assumig tat g is te same for protos ad eutros. Sice for eutros g it is differet from tat of protos g, terefore eq.(2) becomes [5], Z ( Z) N were, g g, g g g (3) 4.2. Williams Formula Te effect of Pauli correctio appears as if te magitude of te excitatio eergy was lowered. Tis meas tat seeral states are blocked, tus teir cotributio i te amout of te excitatio eergy will (2)

3 decrease. et effect will be as if E becomes [E p, (p,,e)], were p, (p,,e) is Pauli correctio term [6], tat is to use te formula, Were, 1(E,) = State desity of te system defied by te state of P, ad E, for oe-compoet. g = Sigle- particle leel at Fermi eergy. E= Excitatio eergy. p, = Pauli blockig eergy for oe- compoet system. p, = Correctio term for state desity formula. P = Particle excito umber for oe-compoet. = Hole excito umber for oe-compoet. = Excito umber. =p +. were(e- p, ) is te Heaiside step fuctio defied as, ( E p, 0 ) 1 E E p, p, d te correctio term p, is gie as [6], p, p( p 1) ( 1) 2g 0 0 For oe-compoet. For two-compoet it is gie as, 1 g g ( E p ),, p, (, E) ( E p,, p, p!! p!! ( 1)! 2 p,, p, Were, p ( p 1) ( 2g 1) p ( p 1) ( 2g ) 1) 2(E,)= State desity of te system defied by te state of P, ad E, for two-compoet. (or g ) = Sigle-Particle leel desity for proto (or eutro). (or 1 g ( E p, ) ( E, ) ( E p, p!!( 1)! 1 = Hole excito umber for proto (or eutro) particle for two-compoet. ) (6) (7) (5) (8) (4) (or ) = Particle excito umber for protos (or eutros) for two-compoet. = Pauli blockig eergy for two-compoet system Pairig ad Modified Williams' Formula wit Back-Sift Fermi Gas: Correctio due to pairig will also cause a reductio i te excitatio eergy E by te amout of eergy, tat is geerated from pairig effect. So, te Effectie Excitatio Eergy, E eff, of te system will be, E eff =E-, te te geeral form, E eff =E- f( ) Were, f() is a correctio fuctio of eergy tat depeds i oe way or aoter o pairig, wic is gie for eergy back-sift. Usig te ESM approximatio, te 1 (,E) of a system will be gie as [3] g E P( ) Bp, (, E, P) ( E P( ) Bp, p!! ( 1)! 1 1 Were, P( )= g ), B p, te modified Pauli eergy ad te eergy gap is gie as [3], d E pase is te pairig eergy for pase trasitio [3] defied as followig, (9) DOI: / Page

4 (, E, 1 K g ) E K ( p, ) p!!( 1)! 1 Nuclear Leel Desity Calculatio of stropysical Reactios. Te simplest Back-Sift Femi Gas model (BSFG) equatio for oe-compoet is more appropriate to iclude te Heaiside step fuctio so tat [3], g E P( ) B S 1(, E, S) ( E P( ) B S) p!! ( 1)! were S is te Back-Sift eergy. 1 V. Results ad Discussios Nuclear state desity is calculated for te uclear reactio iside te core of tree differet mai-sequece stars, amely: Su, Sirius ad Vega. I Table (1) te geeral specificatios of tese stars are listed. (10) (11) Star Su Mass M s * 1 ge x10 9 yr 4.57 Vega 2.1 ~ 1 Sirius * 1 M s =1.999x10 30 g ** 1 R s =1.391x10 6 km Table (1).Geeral specificatios of selected stars Distace Diameter** bs. Mag. Temp. K L (aerage) R s 1 U * 25.0 ly ly 1.93 Lum isual sus isual isual sus isual Spectrum G2V 0Va 1Vm Color Idex Yellow 0.00 Wite 0.02 Wite+ It ca be see from Table (1) tat bot teir ages are less ta te Su ad ae more surface temperature..etc. I tis researc te attetio will be focused o state desity calculatios for reactios usig te ecessary elemets i te selected stars, amely, te uclei of He 4, C 12, N 14, ad O 16. For compariso purpose, te state desity of Fe 54 ucleus will also be icluded. Oly few state desity correctios will be used. I all calculatios made below, te ESM approximatio is assumed. 4 Table (2). Iput alues of state desity calculatios for C 2 He, 6 E max 100 MeV max 4 to 6 oe-compoet (p, ) (3,2) two-compoet (p π, π,p ν, ν) (3,2,0,0) or (3,1,0,1) Sigle particle desity, g /13 (MeV -1 ) Back-sift eergy, S 5 MeV Pairig correctio, U p 3.5 MeV Proto: 8 MeV Nucleo s bidig eergy, B Neuro: 10 MeV Fermi Eergy, F Proto:38 MeV Neutro: 40 MeV Potetial dept, V 38 MeV 14 N , 8 O 54 8 ad 26 Fe 28 uclei. 5.1 State desity calculatio wit o correctios, Ericso's formula Results of oe-compoet ad two-compoet uclear state desity calculated usig Ericso's formula (wit o correctios) are sow i Fig.(1) te oe-compoet results are sow for = 4, 12, 14, 16 ad 54; correspodig to uclei He 4, C 12, N 14, O 16 ad Fe 54 respectiely. Te cofiguratio was fixed at (p,)=(3,2). Te results clearly sow tat as icreases, te alue of g will also icrease ad te state desity also icrease. Te results for two-compoet wit cofiguratio ( =(3,2,0,0) are sow i Fig.(2). DOI: / Page

5 Fig.(1). Te results of te state desity ω 1 as a fuctio of excitatio eergy, for oe-compoet usig Ericso's formula. Te cofiguratio was p,=(3,2) ad = (4, 12, 14, 16, 54) Fig.(2). Te results of te state desity ω 2 as a fuctio of excitatio eergy, for two-compoet usig Ericso's formula 5.2. State desity calculatios wit Pauli correctio, Williams' Formula Te results i Fig.(3) are of state desity for oe-compoet tat did ot use Heaiside formula of eq.(5). Terefore it is see tat tere is a sudde drop i te state desity at lower eergies. t eergy E~ 10 MeV te lie dropped for =4, ad as icrease te drop poit is sifted toward lower eergies reacig ~ 1 MeV at =54. Tis beaior was ot see i oter work suc as [3]. Te same calculatio was made but for two-compoet i Fig.(4). importat remark ere is tat te beaior dropped at iger eergies ta i oecompoet results. For = 4, te lie dropped at E~18 MeV ad at =54 at E~ 5 MeV. Te state desity is calculated based o eq. (4) ad (6), wit te Heaiside fuctio, ad te results are sow i Fig.(5) for oe compoet, ad i Fig.(6) for two-compoet respectiely. Te state desity is was smooter wit icreasig eergy ta before ad tere was o sudde drop i te results. Te Heaiside fuctio actually cut te results lower ta Pauli blockig p eergy ad forced te state desity to start wit alues iger ta tis. Fig.(3). Te results of oe-compoet state desity as a fuctio of Eergy based o Williams' formula witout Heaiside fuctio. Te cofiguratio is (3,2). DOI: / Page

6 Fig.(4). Te same as Fig.(3) for two-compoet system based o Williams' formula witout Heaiside fuctio. Fig.(5). Te results of oe-compoet state desity as a fuctio of Eergy based o Williams' formula wit Heaiside fuctio. Te cofiguratio is (3,2). Fig.(6). Te same as Fig.(5) for two-compoet based o Williams' formula wit Heaiside fuctio. Te cofiguratio is (3,2,0,0). Furtermore, a compariso betwee Ericso's ad Williams' formulae is made by meas of percetage differece. Te formula used is: DOI: / Page

7 for bot oe-compoet ad two-compoet. Tis calculatio was made i order to study te effect of addig Pauli eergy ad te Heaiside fuctio o bot types of calculatios. Te results are sow i Fig.(7) for oecompoet, ad i Fig.(8) for two-compoet; wit similar iput as before. Fig.(7). percetage compariso betwee oe-compoet state desity for Ericso ad Williams formulae. ll iput alues were te same. Fig.(8). percetage compariso betwee two-compoet state desity for Ericso ad Williams formulae. ll iput alues were te same. 5.3 Parig correctio: Te calculated state desity as a fuctio of E usig te pairig formula eq.(9) for te oe-compoet cofiguratio (3,2) is sow i Fig.(9), for te selected rage of mass umbers. Cofiguratio was (3,2) i bot cases. Tis effect appears at low eergy rage ad decreases at iger eergies, because te pairig effect represets te resistace to te excitatio process. It ca be see tat as te mass umber icreases, te pairig effect decreases. Te maximum effect was at =4, te elium ucleus. lso from Fig.(9) tere is a cut-off i state desity for all cases, specially for elium ucleus ad as icrease tis cut-off decreases. Te pairig effect o te state desity calculatios wic are based o eq.(10) is sow i Fig.(10) wit back-sift ad o oter modificatio. From comparig Fig.(9) wit Fig.(10) it ca be see tat at =4, tere was a large cage i te state desity, were it started from E~60 MeV. Tis is because of te back-sift of eergy wic reduces te state desity at lower eergies. Suc beaior may preet te elium ucleus from goig troug fusio reactio due it's iger bidig eergy. DOI: / Page

8 Fig.(9). Te calculated state desity as a fuctio of E for oe-compoet system wit parig correctio. No oter modificatio was used eq.(9). Fig.(10). Te calculated state desity as a fuctio of E for oe-compoet system wit parig correctio ad o back-sift eergy. Modified correctio eq.(10). I Fig.(11) te pairig effect wit BSFG is sow from eq.(11). I tis case te effect of back sift eergy S will furter reduce te eergy sare by eac excito state, terefore te oerall alues will decrease more ta i te preious cases. Fig.(11). Te calculated state desity as a fuctio of E for oe-compoet system wit parig correctio ad back-sift eergy eq.(11). Te effects of S ad, B o state desity for oe-compoet sow i Figs.(9) to (11). Te state desity results cage wit te alue of B were as it icreases te state desity icreases wit fixig eerytig else durig te calculatios. For large alues of bidig eergy B, te state desity results are almost idetical except at te ed, were a sligt reductio occurred. Howeer at lower B alues, tere will be iger cages. Tis DOI: / Page

9 idicates tat state desity calculatios at B is more close to realistic situatio ad tat is ery sesitie to tis alue. lso at B=8 MeV, te result will be ery differet from te alues aboe 10 MeV, ad B=2 MeV as well. Furtermore, i Fig.(12) te calculatios were made for te cofiguratio p==2. It is preferred to use p= for cases of closed-sell uclei as metioed i [3] because i te case of closed-sell uclei tere is a iger cace for bot particle ad ole similarities Fig.(12). Te calculated state desity as a fuctio of E for oe compoet system wit parig correctio. No oter modificatio was used eq.(9).cofiguratio was p==2. calculatio is made of state desity based o parig wit William s formula for same excito umber ad for te same ucleus He 4 i Fig.(13). Tis figure is made i order to sow te total effect of pairig o Williams' formula, based o eq.(11). Te cofiguratio was assumed to be p= wit alues from 1 ad 2. lso te summatio of te state desities is sow. Te summatio of state desities plays importat role i calculatig te reactio cross-sectio [3]. Fig.(13). Te calculated state desity as a fuctio of Eergy for oe-compoet system wit parig correctio ad back-sift eergy. Te cofiguratio p= is used for cases 1 ad 2. Te summatio is also gie. Te set of figures from Fig.(9) to Fig.(13) all used arious pysical parameters. Tese parameters are calculated witi te program codes for eac case. 5.4 State Desity wit 3-D Plots I order to represet te beaior of state desity wit differet excito umbers, for a rage of eergy E, tree-dimesioal plots were made for Ericso ad Williams' formulae below. I te case of Williams' formula te two-compoet results are less ta te oe-compoet results because te Pauli correctio term becomes less, as sow i Fig.(14). DOI: / Page

10 Fig.(14). Tree-dimesioal plot of te state desity based o two-compoet Williams' formula. fixed alue of =4 was used i tis figure. VI. Coclusios Te followig coclusios ca be made: 1. From Ericso formula study of te state desity, it was cofirmed tat ay cage i te cofiguratio will result differet alues of state desity ad, te results sowed tat te depedece o eergy is ot liear. lso i te state desity wit Ericso formula, te alues of iger desities were foud at iger eergies, but te cage was less. 2. lso from Williams' a importat remark was see from tat te cofiguratios (3,2,0,0) ad (3,2), altoug were te same wit proto particles ad oles, but tey resulted i differet beaior at te same eergies. Tis is because of Pauli correctio wic blocks some of te eergy for te excito. 3. Compariso betwee Ericso ad Williams formulae soes tat a) State desities for oe-compoet are geerally iger ta tose of two-compoets for te same cofiguratio ad oter iput alues, b) Ericso's formula gae i iger state desity alues ta Williams' formula, c) s te eergy icreases te differeces icrease betwee oe-compoet ad two-compoet, ad te differeces also icrease betwee Ericso ad Williams formulae, d) s te mass umber icreases, te percetage ratio decreases i faor of Ericso formula. Tere was a otable beaior of two-compoet state desity for Fe ucleus were its alues wet iger ta =12 16 at eergies aboe ~ 90 MeV. Tis idicates tat tis ucleus as more reactio capability at iger eergies. 4- Te oerall compariso wit excito umber ad eergy sowed tat most probable reactio will be at iger eergies ad iger excito umbers. 5- Pairig results also sowed tat selectig te case of p==2 te state desity was wit lower alues ta i te case p=3, =2. Te total excito umber also decreases wic leads to less state desity. Tis was because of te distributio of eergy o less compoets wic gies larger sare for eac particle or ole, tus te state desity will icrease i te case of differet p ad. 6- geeral remark tat was reaced is tat: less state desity meas less reactio probability. Refereces [1]. Burbidge, E. M.; Burbidge, G. R.; Fowler, W..; Hoyle, F. (1957)."Sytesis of te Elemets i Stars". Reiews of Moder Pysics 29 (4): [2]. J. J. Griffi, Pys. Re. Lett. 17(1966)478. [3] Selma, " Neutro Iduced Preequilibrium Nuclear Reactios Usig te Excito Model", P.D. Tesis, Ui. Bagdad, (2009). [4]. C. K. Clie ad M. Bla, Nucl. Pys. 172(1971)225. [5]. E. Bĕták ad P. E. Hodgso, Particle-Hole State Desity i Pre-Equilibrium Nuclear Reactios, Uiersity of Oxford, Report referece OUNP (1998). [6]. F. C. Williams, Jr., Nucl. Pys. 166(1971)231. [7]. rould, M.; Goriely, S. (2003). "Te p-process of Stellar Nucleosytesis: stropysics ad Nuclear Pysics Status". Pysics Reports 384 (1 2): DOI: / Page