Nuclear reactions of heavy ions

Transcription

1 Autho: Facultat de Física, Univesitat de Bacelona, Diagonal 645, Bacelona, Spain. Adviso: Xavie Vinyes Abstact: In this wok nuclea eactions of heavy ions ae studied, focusing on elastic scatteing. A classical and quantum desciption is made in ode to obtain some impotant paametes, such as gazing angle, angula momentum o eaction coss sections, and to compae both desciptions. I. INTRODUCTION. REACTION MECHANISMS The aim of this wok is to study nuclea eactions between heavy ions. We conside heavy ions those with mass numbe geate that cabon s (A=12), although this limit is not always consideed to be the same [1]. The geat inteest of this eactions has many easons; fist, the complex natue of the paticles allows the occuence of many diffeent eactions; besides, fusion eactions may poduce nuclei with high excitation enegy, which allows studying nuclea matte in some special conditions not commonly found in othe ways.[2] In these inteactions diffeent phenomena can take place depending on paametes like the pojectile enegy o the impact paamete. We can distinguish fou inteaction egions: fusion, incomplete fusion and deep inelastic collisions, peipheal and Coulomb egions. When the two ions come vey close to each othe and the incident enegy is high enough, the inteaction leads to the fomation of a compound nucleus; this is a fusion eaction. The odeed motion fom the pojectile and the taget tuns into a caothic themal motion with a cascade of nucleon-nucleon inteactions, a pocess that leads the nucleus to equilibium (themalization). The inteaction between both paticles is descibed by a complex potential fom which we will only conside the eal pat, this is: V eff = Z 1Z 2 e 2 + V e R 0 a + 2 L(L + 1) 2µ 2 (1) whee the fist tem is the Coulomb potential, the second one is the nuclea potential and the thid one is the centifugal tem. We will see late that this potential has a well fo cetain values of the angula momentum below a citical value. If the enegy is highe than the Coulomb baie, the pojectile can get tapped into the well, whee is affected by nuclea inteaction and may poduce fusion. If the enegy is lowe, the pojectile will only be affected by the Coulomb inteaction. When the two ions pass each othe a bit futhe, but nea enough to still allow a stong inteaction, pocesses like deep inelastic and incomplete fusion eactions can take place. Due to this stong inteaction, a consideable faction of kinetic enegy tuns into intenal excitation enegy. In this egion and fo light pojectiles binay fagmentation often occu, leading to pocesses like elastic o inelastic beak-up (whee both fagments come out) o incomplete fusion (whee one of the fagments is emitted while the othe is absobed by the taget, which can be excited). Deep inelastic collisions take place with heavie nuclei, with an incident enegy of about E i 10MeV/nucleon. Fo lage distances between both ions (o lowe enegies) we come into the peipheal egion, whee nucleon tansfe can occu and, fo even lage impact paametes, elastic and inelastic scatteing. In inelastic scatteing eactions, the pojectile inteacts with the taget nucleus tansfeing some enegy to it, so that the taget can get excited. At low incident enegies and fo highly chaged paticles, the excitation of the taget can be due to the Coulomb field (Coulomb excitation); at highe enegies the excitation is due to both the nuclea inteaction and the Coulomb field. In nucleon tansfe eactions one o moe nucleons ae tansfeed fom the pojectile to the taget o vicevesa; these eactions can povide infomation about the stuctue of nuclei. In this wok I will focuse on peipheal eactions and paticulaly on elastic scatteing. Theefoe, a moe detailed explanation of the latte is given below. II. ELASTIC SCATTERING. CLASSICAL DESCRIPTION Elastic collisions ae those that leave unalteed the state of the taget. These will be descibed late though the so called optical potential. Howeve, I will stat giving a bief classical desciption in ode to be able to compae with it the quantum optical potential model.[3] Fom the classical non-elativistic mechanics we can

2 emembe that the fact that the foce is cental ( F ( ) = F ()ˆ) leads to i) the consevation of angula momentum(l = M 2 φ) and ii) the tajectoy of the pojectile is contained on a plane. and φ ae, espectively, the adial and angula coodinates of the pojectile position. The equations of motion can be witten as φ = L/(M 2 ), (2) M = F () + The total enegy can be expessed as: E = T + V = 1 2 Mṙ2 + M 3 (3) 2M 2 + V () Which, consideing (2), can be ewitten as: E V = 2M 2 + 2M 4 ( d dφ )2 fom whee we can isolate dφ and obtain: L/ 2 dφ = ± 2m(E V ) / d. 2 Using the condition of closest appoach d/dφ 0 = 0 ( 0 is the distance of closest appoach) and consideing that 2α + ϑ = π (whee ϑ is the deflection angle and α is the angle between the position vecto at 0 and the asymptote of the outgoing tajectoy), this gives the classical tajectoy fo a paticle in a foce field affected by the inteaction potential V(): L/ 2 ϑ(l) = π 2 d (4) 0 2M(E V ()) /2 If we conside only Coulomb inteaction, the potencial V () on on (4) is V () = Z 1Z 2 e 2 and the integal can be solved analytically, giving: ϑ(l) = 2 actan( MZ 1Z 2 e 2 (5) L 2ME ) (6) Howeve, fo a moe geneal potential that takes into account the nuclea inteaction we cannot obtain an analytical solution and need to use numeical methods. We can conside the scatteing by a taget nucleus of a beam of paticles, all of them with the same mass M and enegy E and each of them chaacteized by its angula momentum L = bp (b is the impact paamete and p = Mv the linea momentum). Then, the numbe of scatteed paticles with angle between θ and θ +dθ (scatteing angle espect to the pola axis) is the same that the numbe of paticles that coss a plane pependicula to the pola axis with impact paamete between b and b + db. Since the scatteing coss-section is defined as the numbe of scatteed paticles pe unit time fo unit incident intensity, we can wite: J dσ 2π sin θdθ = J2πbdb dω whee J is the cuent density of the beam. Fom hee we can obtain the classical diffeential coss section: dσ dω = b sin θ db dθ = L p 2 sin θ dl dθ (7) The scatteing angle θ obseved in expeiments (0 < θ < π) should be distinguished fom the deflection angle ϑ, which is the pola angle of the asymptote of the outgoing tajectoy, consideing that the paticle may plunge o obit aound the cente of foce and theefoe ϑ can be negative o geate than 2π. Both angles ae elated by θ = ±ϑ 2πn with intege n such that 0 < θ < π. Eq. (7) is valid only in the case that L(ϑ) is a single-valued function. If it is multi-valued, the diffeential coss section takes the fom: dσ dω = 1 2p 2 sin θ i d dϑ L=L i (8) whee we have consideed the contibutions fom all the banches of L(ϑ). We conside now a epulsive Coulombian inteaction (5). Using the esult fom Eq. (8) and the expession ϑ(l) fo the Coulomb potential (6) we obtain the diffeential coss section of Ruthefod [4]: dσ R dω = (Z 1Z 2 e 2 4E )2 1 sin 4 (θ/2) (9) We can see that does not depend on the signs of the chages. In the case of a Coulomb potential, the diffeential coss section obtained fom non-elativistic quantum mechanics gives an identical esult. Integating ove all angles we obtain the total coss section (defined as the nombe of paticles scatteed in all diections pe unit time fo unit incident intensity): σ R = ( Z 1Z 2 e 2 1 Mv 2 E )2 [ 1 cos θ ]1 1 which is infinite. This is because of the infinite ange of the Coulomb inteaction; actually σ will be infinite fo any scatteing field diffeent fom zeo at any distance and will only be finite if the field has a cut. In quantum mechanics, howeve, potentials that tend to zeo faste Teball de Fi de Gau 2 Bacelona, June 2016

3 than 1/ 2 have finite total coss-sections. If we add an atactive field (nuclea inteaction) to the Coulombian field, ϑ(l) gets modified and so the tajectoies of the pojectile [5]. As we educe the impact paamete, the nuclea attactive field stats to modify the obit; the limit case when this happens coesponds to the gazing tajectoy. Fo lowe b s it is possible to achieve a balance between nuclea and coulomb foces and have an obiting tajectoy. Fo even smalle b s, the nuclea inteaction can be stonge than the epulsive foce so that the obit plunges, going out with negative deflection angle. III. ELASTIC SCATTERING. DESCRIPTION BY THE OPTICAL MODEL POTENTIAL The optical model consists on teating the scatteing and absoption of nucleons by a nucleus in a simila way than scatteing and absoption of light. As in optics, whee a complex efaction index is used, fo the nuclea eactions we can define a complex potential. This is done because a eal potential would only explain scatteing of the incident paticles, but would not explain thei emoval fom the elastic channel by inelastic pocesses, pe-equilibium eactions and compound nucleus eactions. Theefoe, the imaginay pat of the potential is the one that takes away flux of paticles fom the elastic channel. In thee dimensions we can solve the Schödinge equation to find the elastic scatteing diffeential coss section. This is done using the quantum scatteing fomalism of patial waves [6]. Fom the quantum scatteing theoy we obtain the wave function Ψ: Ψ = 1 2ik (2L + 1)P L (cos θ)(e 2iδ L e ik e ik ) L V C = Z 1Z 2 e 2 (3 2 2R C RC 2 ) R C V C = Z 1Z 2 e 2 R C (11) Typical paametes of the optical potential ae: U 50MeV, W 10MeV, 0 1.2fm, a 0.65fm, V so 4MeV. IV. RESULTS In this section I will pesent the esults obtained fom some calculations using the fotan pogams taj hi1.f, taj hi2.f and taj hi3.f [7], based on the classical theoy, and the pogam nvgopthi.f [8], which uses the quantum theoy of the optical potential model. Some magnitudes of inteest in elastic scatteing will be shown, such as the effective potential, the distance of closest appoach in tems of the angula momentum, classical tajectoies, the values of L and θ coesponding to the gazing tajectoy and finally I will show some diffeential coss section values fo diffeent eactions in ode to compae the classical with the optical model esults. A. Effective Potential and 0(L) As explained above, the effective potential between two heavy nuclei includes the Coulomb and the nuclea inteaction. Figue (1) shows this potential as a funtion of the adial distance between the two nuclei 16 O and 88 S fo an incident enegy of E LAB = 60MeV, calculated using the pogam taj hi3.f. The ed line coesponds to an angula momentum L=0 and the geen one coesponds to the gazing tajectoy (the blue hoizontal line is the enegy at the cente of mass fame of efeence, E CM = A2 A 1+A 2 E LAB = MeV ). (whee δ L ae the phase-shifts) and we can find the angula distibution of the scatteed nucleons, the total elastic coss section (integated fo all angles) and the eaction coss section. The total optical potential can be expessed as: V () = V c () + U f u () + iw f w () + V so () (10) The fist tem on (10) is just the Coulomb potential (11). The second tem is the eal pat of the nuclea potential and thid is the imaginay pat; both of which (f u and f w ) have a Saxon-Woods fom. The fouth is the spin-obit tem, which allows calculating the polaization of the scatteed beam, although we won t use this hee. FIG. 1: Effective potential as a function of the adial distance Teball de Fi de Gau 3 Bacelona, June 2016

4 FIG. 2: Distance of closest appoach in tems of the angula momentum fo the eactions 16 O + 88 S and 16 O + 40 Ca. Fig. (2) shows, fo two diffeent eactions, the distance of closest appoach as a function of the angula momentum L. We can see a jump in 0, that coesponds to the angula momentum of the gazing tajectoy, L g. Below, the nuclea inteaction stats to make effect. The distance of closest appoach coesponding to the discontinuity in fig.(2) can be compaed (table I) to the stong inteaction adius R int, which is the distance whee nuclea inteaction stats to be elevant: FIG. 3: Tajectoies of the pojectile 16 O when inteacting with a nucleus of 88 S. Each line coesponds to a diffeent vaule of the angula momentum L. C. Optical model Now we intend to obtain anothe estimation of L g using the optical potential model. To do so we have used the pogam nvgopthi.f, which gives the diffeential coss section as atio to the Ruthefod value in tems of the scatteing angle. The esult of this is shown in fig.(4). R int = R 1 + R fm (R i = 1.12A 1/3 i 0.94A 1/3 i ae the half-density adii). Reaction 0 ± 0.15(fm) Rint(fm) discepancy 16 O + 88 S O + 40 Ca g TABLE I: Compaison between the distance of closest appoach coesponding to L g and the stong inteaction distance R int. We can see that both values ae quite simila, with a discepancy within two times the eo. B. Tajectoies Fo the eaction 16 O + 88 S I have chosen a set of L values aound the appoximate value of L g obtained fom fig.(2) (fom 27 to 28.5 with intevals of 0.1, in units of ), and dawn some of thei tajectoies using the data obtained by unning taj hi1.f. In fig.(3) we can see, in black colou, the gazing tajectoy coesponding to the angula momentum L classical g = (28.0 ± 0.1) The tajectoies above suffe basically only the Coulomb inteaction and the ones beneath ae affected by the nuclea inteaction. FIG. 4: σ/σ R as a function of θ. The black line shows a simple classical appoach. Fo θ > θ g, σ fo elastic scatteing goes to zeo (the nuclei inteact so stongly that an elastic collision cannot take place), while fo θ < θ g it takes the σ R value. In ed is shown the optical model esult, that pesents some oscillations and falls appoximately exponentially. The angle θ g of the gazing tajectoy can be found consideing that θ g = θ 1/4, whee the quate-point angle θ 1/4 is the one that coesponds to σ/σ R = 0.25 [9]. Fom the data obtained fom numeical calculations (file nvgopthi.dat ) we get: θ g = (82 ± 1) o that coesponds to σ/σ R = We can obtain the Teball de Fi de Gau 4 Bacelona, June 2016

5 gazing angula momentum using: L opt g = n cot( θ 1/4 2 ) whee n = Z1Z2e2 v is the Sommefeld paamete (v is the pojectile initial velocity). Fo the eaction of 16 O+ 88 S, with initial kinetic enegy E LAB = 60MeV, and consideing the units e2 c = 1 137, we obtain: L opt g = (28.6 ± 1.2) Compaing this esult with that obtained on section B (tajectoies of the classical model) we can see that the esults ae compatible, since d < 2 δ(l), and the elative discepancy is d = L classical L optical L classical = % D. Reaction coss section Reaction coss sections have been calculated fo the classical and optical methods fo some eactions of 16 O with diffeent tagets [10]. The esults obtained ae shown in Table (II). Taget σ classical (fm 2 ) σ optical (fm 2 ) elative discepancy 92 Mo Z Z S S F e C T i Ca Ca Ni Ni Ni Ni TABLE II: Reaction coss section values (integated ove all angles) fo the inteaction of 16 O with the diffeent tagets in the fist column. We can see that the discepancy between the two methods is not elevant; in othe wods, the classical desciption gives a vey good apoximation fo the coss section with the consideed eactions. V. CONCLUSIONS In fig.(2) we can see that fo the 40 Ca taget (Z = 20) the L g value is geate than fo the 88 S taget (Z = 38). This is because fo smalle atomic numbe, the Coulomb inteaction is weake (see (5)) and theefoe the limit value fo angula momentum in which nuclea inteaction stats to be elevant inceases. Consideing the esults obtained eithe fo the gazing angula momentum and fo the eaction coss sections, we can conclude that the diffeence between the classical desciption and the optical model is not vey significant. Even though the nuclea inteactions teated occu in a scale in which it would seem that quantum effects ae dominant, the heavy ions we conside have shot wavelenghts and often lage angula momentum, which implies that the inteactions can be descibed as classical paticles that move along a localized tajectoy (fig.(3)). When quantum effects (like intefeence o diffaction) become elevant, the classical theoy can be modified to take them into account (this is called semi-classical appoximation). Acknowledgments I would like to thank fist of all my adviso Xavie Vinyes, fo his useful help and kindness; and also my dea fiends Raimon Luna and Cales Riveo, fo thei infinite patience and suppot. [1] In some cases the α paticle is consideed the bodeline fo heavy ions [2] Hodgson, Gadioli. Intoductoy nuclea physics, Oxfod science publications, Chapte 23. [3] Salvat. Notes Elastic collision of chaged paticles with atoms [4] Deduced in 1911 fo scatteing of α paticles by atomic nuclei.it descibes suitably the angula distibution of scatteed paticles in many cases, except when the two ions ae equal [5] See: G.R.Satchle. Intoduction to Nuclea Reactions. MacMillan, London pages [6] Hodgson, Gadioli. Op. cit. Chaptes 13, 20 [7] F. Salvat, taj hi.f, Unpublished [8] P.E.Hodgson, nvgopthi.f, Oxfod Univesity Repot. [9] Reine Bass. Nuclea Reactions with Heavy Ions. Chapte 1. Spinge-Velag. Belin 1980 [10] F.D. Becchetti. Elastic and inelastic scatteing of 16 O and 12 C fom A = 40 96nuclei. Nuclea physics, Noth- Holland Publishing Co., Amstedam Teball de Fi de Gau 5 Bacelona, June 2016