Differential Cross Section of Elastic and Inelastic p 15 N Scattering

Transcription

1 NUCLEAR THEORY, Vol. 32 (2013) eds. A.I. Geogieva, N. Minkov, Heon Pess, Sofia Dfeential Coss Section of Elastic and Inelastic p 15 N Scatteing E. Ibaeva 1, N. Butebaev 1, M. Zhusupov 2 1 Institute of Nuclea Physics, Almaty, Kazakhstan 2 Al-Faabi Kazakh National Univesity, Almaty, Kazakhstan Abstact. The calculation of dfeential coss sections fo elastic poton scatteing at enegies 0.6 and 1.0 GeV and inelastic poton scatteing (at the level J =5/2 + ) at enegies 0.415, 0.8 and 1.0 GeV fom 15 N nucleus at E = 0.4, 0.8 and 1.0 GeV was made within the Glaube s dfaction theoy. It is shown that in the appoximation of double scatteing using the shell wave function 15 N the amplitude of 15 N- pocess can be calculated analytically. The calculation of dfeential coss section in the optical limit, allows us taking into account the collisions with nucleons on dfeent shells. 1 Intoduction 15 N nucleus is stable, the numbe of its neutons is just by one moe than the numbe of the potons. Though its abundance in natue is just 0.36%, it plays an impotant ole in CNO-cycle, being a basis of the nucleosynthesis of 12 C, 16 Oand 4 He. Theefoe, in liteatue, the pocesses involving 15 N ae mainly eviewed at low (astophysical) enegies [1]. The vaious chaacteistics of the 15 N(p,γ) 16 O, 15 N(p,α) 12 C eactions ae calculated, which fom the banch point of the CNO-cycle [2, 3]. Thus, definition of the eaction coelation paametes is necessay in ode to simulate the pocess of enegy poduction in stas, and nucleosynthesis of cabon, nitogen and oxygen isotopes in the pocess of hydogen buning in stas. Hee we conside the elastic and inelastic scatteing of potons fom 15 N nucleus at highe enegies (fom 0.4 to 1.0 GeV). Theefoe we use the Glaube dfaction theoy [4], which descibes the poton scatteing at the intemediate enegies in the most appopiate way. The Glaube theoy is attactive because it allows us to sepaate the stuctual (dependingon the wave function (WF) of the taget nucleus) and dynamic (depending on the opeato of multiple scatteing) components of the scatteing amplitude. The input paametes of the theoy ae the WF of the taget and the elementay nucleon-nucleon amplitude. Paametes of the elementay nucleon-nucleon amplitudes ae taken fom the expeiments by pp and pn scatteing [5 7]. Cuently, when all pecise calculations ae made by the numeical method with the help of lage compute softwae, we have made the analytical calculation of the matix element with the WF of 15 N in the shell model and the opeato, whee the fist and the second ode of collisions ae taken into account. With 170

2 Dfeential Coss Section of Elastic and Inelastic p 15 N Scatteing this appoach, we wee able to show the contibution to the dfeential coss section (DCS) by dfeent odes of collisions and to show the contibution to the DCS fom the inteio egion of the nucleus and the nuclea peiphey. One of the aim of the pesent pape to calculate the DCS of poton scatteing on 15 N nucleus in the optical limit of the Glaube dfaction theoy (when only single collisions ae taken into account in the opeato of multiple scatteing) and to analyses this coss section sensitivity in poton scatteing on the nucleons of dfeent shells. Theefoe to descibe the intenal stuctue of 15 N, we use the shell model [8]. It is known that the multiple scatteing seies conveges apidly and each next tem of the seies contibutes to the coss section by seveal odes less than the pevious one. The optical limit appoximation popely descibes the DCS at the font angles. 2 Bief Fomalism The scatteing matix element (amplitude) within the Glaube theoy is witten as follows [4]: M = ik d ρdr 2π A exp (i q ρ)δ( R A ) Ψ J M j f Ω Ψ JMj i, (1) M j M j whee Ψ JMj i and Ψ J M j f ae the WFs of initial and final states, ρ is the impact paamete, pependicula to the diection of the pojection; A is the numbe of nucleons in a taget, R A = 1 A A n=1 n is the coodinate of the nucleus mass cente, q = k k is the tansfeed momentum, k, k ae the momenta of the pojectile and the scatteed paticles in the cente-of-mass system, Ψ J M j f Ω Ψ JMj i is the matix element of the tansfe fom the initial to the final states. The gound state of 15 N nucleus is the level of negative paity with J π = 1/2,T =1/2and (1s) 4 (1p) 11 configuation. The shell WF be epesented in the fom [8] Ψ i,f ( i )=Ψ n0,l 0,m 0 ( 1,..., 4 )Ψ n1,l 1,m 1 ( 5,..., 15 ) =Ψ 000 ( 1,..., 4 )Ψ 11m1 ( 5,..., 15 ), (2) whee Ψ n,l,m ( 1, 2,...) = ν Ψ n,l,m( ν ) is the poduct of single-paticle functions, ν ae the single-paticle coodinates of nucleons. The excited state of 15 N nucleus is the level of positive paity with J π = 5/2 + and (1s) 4 (1p) 10 (1d) 1 configuation [8] Ψ f ( 1, 2,...)= (1s) 4 (1p) 10 (1d) 1 = m 1m 2 Ψ 000 ( 1,..., 4 )Ψ 11m1 ( 5,..., 14 )Ψ 22m2 ( 15 ). (3) 171

3 E. Ibaeva, N. Butebaev, M. Zhusupov Opeato Ω in the Glaube theoy is witten in the fom of a multiple-scatteing seies Ω=1 = A A 1 ω ν ( ρ ρ ν ) ω ν ω τ + ω ν ω τ ω η + +( 1) A 1 ω 1 ω 2 ω A, (4) ω ν ν<τ ν<τ<η whee the fist tem conside the single collisions, the second tem is the double collisions and so on, up to the last tem which conside A-multiple collision. Sepaate pofile functions ω ν ae expessed though the elementay pn-amplitudes f pn (q): ω ν ( ρ ρ ν )= 1 d q ν exp ( i q ν ( ρ ρ ν ))f pn (q ν ). (5) 2πik The elementay amplitude itself is witten in the standad manne f pn (q ν )= kσ pn 4π (i + ε pn )exp( βpn 2 qν/2), 2 (6) whee σ pn is the total poton scatteing coss section by the nucleon, ε pn is the atio of the eal pat of the amplitude to the imaginay one, and β pn is the slope paamete at the amplitude con ae taken fom [5 7]. To announce the key items of the deivation. 1. In the expession (4) we ae esticted only the fist two tems, owing to that fact, that each next tem gives a contibution to the coss section smalle by seveal odes of magnitude than the pevious one. 2. Substituting the seies of multiple scatteing (4) in the amplitude (1), and then integating it with the impact paamete d ρ and momentum d q ν leads to the following esult: whee Ω= 2π ik f pn (q) i<j=1 ω i = 15 ω i ω j = ( 2π ( q ) ) 2 ω i ik f pn ω i ω j, (7) 2 i<j=1 exp (i q ρ i ), i<j=1 ( exp i q ) 2 ( ρ i + ρ j ) δ( ρ i ρ j ). 3. We epesent the opeatos of single and double collisions as the sum of opeatos affecting the nucleons at dfeent shells: 4 14 ω i = ω i + ω i + ω 15, (9) 172 i=5 (8)

4 Dfeential Coss Section of Elastic and Inelastic p 15 N Scatteing i<j=1 ω i ω j = 14 i<j=1 ω i ω j + ω i ω 15. (10) 4. Replacing the flat vectos ρ i (on which they depend ω) with thee-dimentional i, 15, expending ω =exp(i q ) in the Bessel seies: exp (i q ) =4π λ=0 μ= λ λ π (i) λ 2q J λ+1/2(q)y λμ (Ω ) Y λμ (Ω q ), and epesenting the WF in the centally symmetic field is factoized into adial R nl ( ν ) and angula Y lm (Ω k ) pats: Ψ nlm ( i )=R nl ( i )Y lm (Ω ),thenitis possible to integate the matix element (1) in the spheical system of coodinates. Detailed desciption of the elastic p 15 N scatteing may be found in [9], hee we show the esult fo the inelastic scatteing. The matix element of single scatteing: M (1) ( q) = k k 2πf pn (q) λμ whee R 22 () 1 J λ+1/2 (q) R 11 () π = 2q (i) λ R 22 () 1 J λ+1/2 (q) R 11 () Y 2m2 (Ω ) Y λμ (Ω ) Y 1m Y λμ (Ω q ), (11) Y 2m2 (Ω ) Y λμ (Ω ) Y 1m Y λμ (Ω q ) λμ 0 R 22()R 11 ()J λ+1/2 (q) 3/2 d, (12) = 5(2λ +1) 3 4π λ λμ1m 2m 2. (13) λμ The matix element of double scatteing: M (2) ( q/2) = M (2) sd ( q/2) + M (2) pd ( q/2), (14) the uppe indexes indicate the nucleons of the shell whee collision occus. M (2) sd ( q/2) = = C(q/2) (i) λ R 00 ()R 22 () 1 J λ+1/2 (q) R 00 ()R 11 () λμ Y 00 (Ω )Y 2m2 (Ω ) Y λμ (Ω ) Y 00 (Ω )Y 1m (Ω ) Y λμ (Ω q ), (15) 173

5 E. Ibaeva, N. Butebaev, M. Zhusupov whee C(q/2) = ik π ( (ik) 2 6(2π)2 q/2 f 2 q ), (16) 2 R00 ()R 22 () 1 J λ+1/2 (q) R 00 ()R 11 () = 0 R 00 () 2 R 22()R 11 ()J λ+1/2 (q) 3/2 d, (17) Y 00 (Ω )Y 2m2 (Ω ) Y λμ (Ω ) Y 00 (Ω )Y 1m (Ω ) = 1 Y2m 4π 2 (Ω )Yλμ (Ω )Y1m (Ω )dω. (18) The last integal is analog of (13). M (2) pd ( q/2) = = C(q/2) (i) λ R 11 ()R 22 () 1 J λ+1/2 (q) R 11 ()R 11 () λμ Y 1m1 (Ω )Y 2m2 (Ω ) Y λμ (Ω ) Y 1m (Ω )Y 1m (Ω ) Y λμ (Ω q ), (19) C(q/2) = 15 2 C(q/2), R 11 ()R 22 () 1 J λ+1/2 (q) R 11 ()R 11 () = 0 R 11 () 2 R 22 ()R 11()J λ+1/2 (q) 3/2 d, (20) Y 1m1 (Ω )Y 2m2 (Ω ) Y λμ (Ω ) Y 1m (Ω )Y 1m (Ω ) Y λμ (Ω q ) = 15(2λ +1) ( 1) m2 λ020 L 0 L 010 L0 L (4π) 3/2 LML M λμ2 m 2 L M L M 1m LM LM1m 1m 1 Y λμ (Ω q ), (21) It is impotant to note that in appoximation of two multiple collisions all integals ae taken analytically and, theefoe thee is no loss of pecision inheent fo numeical integation. The DCS is detemined by the squaed modulus of the espective matix elements as 174 dσ dω = 1 2J +1 M (1) (2) ( q) M ( q) 2. (22)

6 Dfeential Coss Section of Elastic and Inelastic p 15 N Scatteing In the optical limit, the fist tem of the multiple scatteing seies (7) emains only in opeato Ω, being the sum of single collisions with all nucleons A Ω= ω ν, (23) Let s divide (23) into tems coesponding to the scatteing ove s- and -shells: 4 ω ν = ω ν + ω ν, (24) Afte substituting the opeato (24) into (1), we obtain the matix element as a sum of two components M (1) ( q) = k [ ] k f pn (q) M s (1) ( q)+m p (1) ( q). (25) ν=5 whee M s (1) ( q) = M p (1) ( q) = m Ψ 000 ( ν ) 15 Ψ 11m ( ν ) ν=5 4 d ν, (26) ω ν 2 15 ω ν ν=5 ν=5 d ν, (27) M s (1) ( q) is esponsible fo the scatteing on the 1s-shell nucleons; M p (1) ( q) is esponsible fo the scatteing on the 1-shell nucleons. Those matix elements ae the ovelap integals of the WFs and the opeatos ω ν acting ove the coodinates of the nucleons in the coesponding shells. The DCS is detemined by the squaed modulus of the espective matix elements as dσ OL dω = 1 M s (1) ( q)+m (2) p ( q) 2. (28) 2J +1 3 Results and Discussions We have calculated the DCSs fo elastic scatteing at enegies 0.6 and 1.0 GeV and inelastic scatteing at the level J =5/2 + at enegies 0.415, 0.8 and 1.0 GeV. Figue 1 shows the contibutions fom dfeent collision multiplicities to the DCS fo elastic scatteing at enegies 0.6 (a) and 1.0 GeV (b). As it can be seen fom the figues, the single scatteing (dash cuve) gives the main contibution to the fowad angles, but it deceases quickly and then the double scatteing (dotted cuve) begins to dominate at the lage angles. The minima at θ 14 in Figue 1a and at θ 9 in Figue 1b, caused by the destuctive intefeence, appea at pointswhee the patial coss sections is cossed 175

7 E. Ibaeva, N. Butebaev, M. Zhusupov Figue 1. Contibutions fom the dfeent collision multiplicities to the DCS at E =0.6 (a) and 1.0 (b) GeV. Cuves dash, dotted and solid ae the contibutions single, double collisions and thei sum. each othe, because the seies of multiple scatteing ae sign-changing. In all this cases the esulting coss section at small angles is lowe than the patial coss section of single scatteing. The eason of such behavio is the fact that the esulting coss section is equal to the dfeence single and double amplitudes (see Eq. (22)). Additional minima in the patial coss sections at θ 33 in Figue 1a and at θ 23 in Figue 1b aise because of the use of the ealistic WFs that contain the angula pat. As a esult, the amplitude can change sigh. Since the DCS is the squae of the amplitude, it leads to a minimum in the DCS. Figue 2 shows the contibution to the DSC fom the scatteing on the nucleons of 1s (dash cuve) and 1p (dotted cuve) shells in the optical limit (when we 176

8 Dfeential Coss Section of Elastic and Inelastic p 15 N Scatteing Figue 2. Contibution to the single coss-section (solid cuve) fom the scatteing on the nucleons of 1s (dash cuve) and 1p (dotted cuve) shells at E =0.6 GeV (a) and 1.0 GeV (b). Hee, the solid cuve is the same as dash cuve in Figue 1. take into account only single scatteing in seies of multiple scatteing, it is solid cuve). We can see that at small angles the main contibution to the DCS gives 1p-shell nucleons scatteing. With inceasing scatteing angles the contibution fom the 1s-shell nucleons scatteing deceases slowly than the DCS fom the 1p-shell nucleon scatteing. At some angles they become equal. At these points (whee the DCS ae equal) aise the intefeence of amplitudes (constuctive o destuctive depend on sigh of amplitudes). Fo example thee ae the constuctive intefeence at θ 13 and 32 in Figue 2a, and the constuctive intefeence at θ 9 and destuctive one at θ 15 and 28 in Figue 2b. Wheeas, in Figue 2a the deep minima obseved in the DCS on 1p-shell nucleons do not affect the behavio of the esulting coss section. 177

9 E. Ibaeva, N. Butebaev, M. Zhusupov Figue 3. Contibutions fom dfeent collision multiplicities to the DCS of inelastic scatteing at E =0.415 (a), 0.8 (b) and 1.0 GeV (c). Dash, dotted and solid cuves coespond to single, double and thei sum scatteing. Why the scatteing on 1-shell nucleons is dominated at fowad angles and the scatteing on 1s-shell nucleons is dominated at lage angles? This occus due to the fact that the momentum tansfe inceases with an incease of the scatteing angles and poton can penetate deepe into the nucleus and it inteacts with the nucleons of intenal 1s-shell. 178

10 Dfeential Coss Section of Elastic and Inelastic p 15 N Scatteing Let s discuss inelastic scatteing. Figue 3 shows the contibution to inelastic scatteing DCS fom single and double collisions at E =0.415 (a), 0.8 (b), and 1.0 (c) GeV. The DCS at zeo angle tends to zeo because of the othogonality of the initial and the final states WFs. Figue 3a shows that single scatteing dominates in the whole angula egion. This can be explained the fact, that the scatteing of poton (occus at the nuclea peiphey) on the outside nucleon, located on 1d-shell. Double collisions in the fist maximum ae by thee odes of magnitude smalle than the single collisions, and in the esulting coss section they lead to slight filling of the minimum at θ 22 and slight decease the coss section at θ>35.anothe pictues ae shown in Figues 3b,c. In the egion of fowad angles (θ up to 25 ) the single collisions give the main contibution to the esulting DCS, but they apidly decease at θ>25, and the double collisions at θ>30 become compaable with the single ones. At the lage angles they begin to dominate and detemining the behavio of the coss section. This happens because with incease of the enegy the highe ode collisions comes moe signicant contibution to the DCS. This is the esult of the fact, that the moe enegetic colliding paticles can penetate deepe into the inteio of the nucleus and collide with a moe numbe of nucleons. The tansfe momentum in the eaction also inceases with incease of the scatteing angle, so that the contibutions of the collisions of highe multiplicity become athe moe signicant at the lage angles. As in the elastic p 15 N scatteing, the additional minima in the double patial coss sections (at θ 16 and 30 in Figue 3a, at θ 11 and 20 in Figue 3b, at θ 17 and 22 in Figue 3c) aise due to the use ealistic WFs that include an angula pat. As a esult, the amplitude can change sigh and it leads to the minimum in the DCS. 4 Conclusions The dfeential coss sections of the elastic and inelastic (the level J =5/2 + ) p 15 N scatteing wee calculated within the famewok of the Glaube dfaction theoy at intemediate enegies fom to 1.0 GeV. It is show that the application of the Glaube dfaction theoy to the p 15 N scatteing with WF pesented in the fom of Gaussian functions and with the opeato expessed by an exponential function allows us to calculate analytically the scatteing amplitude and take into account the stuctual components of the WFs. It is evealed the single scatteings ae dominated in the egion of fowad angles, while the contibution fom the double scatteings only slightly decease the DCS. At lage angles the double scatteings become dominant and detemine the behavio of the coss section. Such behavio of DCS is intepeted as follows. With the incease of poton enegy, they can penetate deepe into the inteio of the nucleus and can be escatteed on a moe numbe of nucleons. Consideing the dependence of the DCS on scatteing ove nucleons in dfeent shells (in the optical limit) we can see that the small numbe of nucleons 179

11 E. Ibaeva, N. Butebaev, M. Zhusupov on the 1s-shell lead to thei substantially decease of the contibution to the esulting coss section in the fowad angles in compaison to the scatteing on the 1p-shell nucleons. Howeve, at lage angles the patial DCS of scatteing on the nucleons of 1s-shell begins to dominate and the esulting DCS is detemined by the competing contibution of both patial coss sections. 5 Acknowledgements This wok was suppoted by the Gant Pogam of the Ministy of Education and Science of the Republic of Kazakhstan 1124/GF and 0601/GF. Refeences [1] R.J. deboe, P.J. LeBlanc, S. Falahat, et al., Phys Rev. C 85 (2012) [2] A.M. Mukhamedzhanov, M.La Cognata, and V. Koha, Phys Rev. C 83 (2011) [3] P.J. LeBlanc, et al., Phys Rev. C 82 (2010) [4] R. G. Glaube, in Lectues in Theoetical Physics (Intescience, New Yok; London, 1959). [5] L. Ray, Phys Rev. C 19 (1979)1855; Phys Rev. C 20 (1979) [6] J.P. Auge, C. Lazad and R.J. Lombad, J. Phys. G 7 (1981) 1627; J.P. Auge, et al., J. Phys. G 11 (1985) 751. [7] D.K. Hasell, Phys Rev. C 34 (1986) 236. [8] M.A. Zhusupov, et al., Izv. Rus. Akad. Nauk. Fisika 32 (1968) [9] E.T. Ibaeva, et al., Phys.of Atom.Nucl. 73 (2010)