Igor Gontchar, Maria Chushnyakova Omsk, Russia Nucleus 2015

Transcription

1 SYSTEMATIC COMPARISON OF HEAVY-ION FUSION BARRIERS CALCULATED WITHIN THE FRAMEWORK OF THE DOUBLE FOLDING MODEL USING TWO VERSIONS OF NUCLEON-NUCLEON INTERACTION Igor Gontchar, Maria Chushnyakova Omsk, Russia Nucleus 2015

2 1 Experimental fusion excitation function Fig. 1 [1] Newton et al., Phys. Rev. C 64, (2001)

3 2 Theoretical description CCM: Coupled channels model reasonable for the subbarrier cross sections BPM: Single barrier penetration model gives the upper limit of excitation function TMSF: Trajectory model with surface friction accounts for the energy dissipation; moves down the excitation function

4 3 Double folding potentials with M3Y effective NN (nucleon-nucleon) interaction [2-4] Migdal effective NN interaction [5] [2] Satchler, Love, Phys. Rep. 55, 183 (1979) [3] Bertsch et al., Nucl. Phys. A 284 (1977) 399 [4] Anantaraman et al. Nucl. Phys. A 398 (1983) 269 [5] Migdal, Theory of Finite Fermi Systems and Applications to Atomic Nuclei (Interscience, New York, 1967)

5 Comparison of our fluctuation-dissipation model with the experiment for a single reaction 4 Fig. 4 [6] Chushnyakova, Bhattacharya, Gontchar, Phys. Rev. C 90, (2014)

6 M3Y vs experiment (91 points) 5 Fig. 5 r σ σ / σ = (1) th exp

7 M3Y vs experiment (91 points) 5 M3Y =>? Migdal Fig. 5 r σ σ / σ = (1) th exp

8 6 Double folding model with M3Y NN interaction: (, ) = (, ) + (, ) U R E U R E U R E n P nd P ne P U R E g E dr dr r F s r (, ) ( ) ρ ( ) ( ρ ) v ( ) ρ ( ) = nd P P P T AP P v FA D AT T (2) (3) P Fig. 6 Projectile R Target T Density Dependent M3Y 3 ( ) ( ) v ( s) = G [exp s r ] s r D Di vi vi i= 1 (4) [7] Khoa et al., Phys. Rev. C 56, 954 (1997) [8] Gontchar, Chushnyakova, Comp. Phys. Comm. 181 (2010)

9 7 Double folding model with Migdal forces: The quote from Zagrebaev et al., Phys. Elem. Part. At. Nucl. 38, 892 (2007).

10 8 Double folding model with Migdal forces: (5) (6) [5] Migdal, Theory of Finite Fermi Systems and Applications to Atomic Nuclei (Interscience, New York, 1967) [9] Zagrebaev et al., Phys. Elem. Part. At. Nucl. 38, 892 (2007).

11 9 Two versions of the NN potential 16 O+ 208 Pb Fig. 9

12 M3Y vs Migdal: 10 Systematic comparison of the barriers Barrier heights Barrier radii (7) 1/3 1/3 ( ) B = Z Z / A + A (8) Z P T P T Fig. 10

13 Comparison of BPM cross sections with the experimental ones for a single reaction 11 Fig. 11 [1] J. O. Newton et al., Phys. Rev. C 64, (2001)

14 Comparison of BPM cross sections with the experimental ones for a single reaction 12 Fig. 12 [1] J. O. Newton et al., Phys. Rev. C 64, (2001)

15 Migdal vs experiment (91 points) 13 Fig. 13 r σ σ / σ = (1) th exp

16 Conclusions 14 The comparison made for 19 reactions with spherical nuclei revealed that the Migdal barriers are always higher by several percent than the M3Y barriers. This results in rather low fusion cross sections calculated with the Migdal forces: most of the calculated cross sections significantly (up to 40%) underestimates the experimental values. Using the Migdal nucleon nucleon interaction with the constants from Ref. [5] for describing fusion (capture) cross sections is questionable. [5] A. B. Migdal, Theory of Finite Fermi Systems and Applications to Atomic Nuclei (Interscience, New York, 1967).

17 Fig Comparison of the experimental charge density [10] with the calculated one within the Hartree Fock approach with the SKX coefficient set [10] H. de Vries et al., At. Data Nucl. Data Tables 36, 495 (1987)

18 theor expt rms ( ch th ch exp ) ( ch th ch exp ) δ = 2 ρ ρ / ρ + ρ (20)

19 The ratio of the calculated rms charge radius Rq th to the experimental one Rq exp [10]. 12 C 16 O 28 Si 32 S Nucleus R / R q th q exp 36 S 92 Zr 144 Sm 204 Pb 208 Pb [10] H. de Vries et al., At. Data Nucl. Data Tables 36, 495 (1987)

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22 Reactions 12 C+ 92 Zr, 16 O+ 92 Zr, 28 Si+ 92 Zr Expt. data Refs. [1] Newton et al., Phys. Rev. C 64, (2001) 12 C+ 144 Sm [11] Kossakowski et al., Phys.Rev. C 32, 1612 (1985) [12] Mukherjee, Hinde et al., Phys. Rev. C 75, C+ 208 Pb (2007) 12 C+ 204 Pb [13] Berriman, Hinde et al, Nature (London) 413, 144 (2001) 16 O+ 144 Sm [14] Leigh, Dasgupta et al., Phys. Rev. C 52, 3151 (1995) [15] Morton, Berriman et al., Phys. Rev. C 60, O+ 208 Pb (1999) 16 O+ 204 Pb [16] Dasgupta et al, Phys. Rev. Lett. 99, (2007) 28 Si+ 208 Pb, 32 S+ 208 Pb [17] Hinde et al., Nucl. Phys. A 592, 271 (1995). 36 S+ 208 Pb [18] Yanez et al., Phys. Rev. C 82, (2010) 36 S+ 204 Pb [19] Hinde et al., Phys. Rev. C 75, (2007)

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24 2 Fusion (capture) cross section 2 Lmax π σ = + 2mE R c. m. L= 0 ( 2L 1) T L (1) Single Barrier Penetration Model T L 1 = 1 + exp 2 π / ( U E ) ( ω ) BL c. m. BL (2) Fig. 3

25 Woods Saxon profile: 3 Un( R) = VWS 1+ exp 1/3 1/3 ( ) R r A + A WS P T a WS 1 (3) P Projectile R Target Fig. 4 T

26 F U Classical Trajectory Model with the Surface Friction (TMSF) ( ) 2 dp = F + F +Φ dt +ξχ D dt Dq q U cen Dq q t = du dq tot (7) ( ) ( ) Φ = F s Γ t s ds 0 Dq 1 t s Γ( t s) = exp τ τ F (9) cen F L = mq Dq D q 2 2 q 3 q (5) dq = dt (6) (8) 2 pq du n = KR mq dq 2 dun = θkr dq < ξ ( t) >= 0 < ξ ( t ) ξ ( t ) >= Γ( t t ) p m (11) (12) q (10) (13) (14) 4 [2] D.H.E. Gross, H. Kalinowski, Phys. Rep. 45 (1978) 175 [3] P. Fröbrich, Phys. Rep. 116 (1984) 337

27 5 Fusion (capture) cross section 2 Lmax π σ = + 2mE R c. m. L= 0 ( 2L 1) T L (1) TMSF T L Ncap L = (15) N tot Fig. 5

28 Two versions of the NN potential 16 O+ 208 Pb

29 M3Y vs Migdal NN forces Systematic comparison of the fusion barrier parameters: a) height b) radius c) curvature

30 Classical Trajectory Model with the Surface Friction (TMSF) ( ) 2 dp = F + F +Φ dt +ξχ D dt q U cen Dq q p q (5) dq = dt (6) m q 30 dl = F dt + bχ 2D dt Dϕ ϕ (20) L dϕ = m q dt (21) F ( ) 2 L Ls dun Dϕ = Kϕ m q dq (22) D ϕ du K n = θ ϕ dq 2 (23) [2] D.H.E. Gross, H. Kalinowski, Phys. Rep. 45 (1978) 175 [3] P. Fröbrich, Phys. Rep. 116 (1984) 337

31 Parameters of the matter distribution 31 R ch [9] I. Angeli, At. Dat. Nucl. Dat. Tables 87 (2004) 185 ach arbitrary R 0ch 3 2 7π a = Rch ch (20) R = R 0ch 0A (22) aa = ach r 2 < chp > < rchn > 7π N Z (23)

32 7 K R = MeV zs a AT = 0.510fm F Dq p du n = KR mq dq 2 (10) Fig. 6

33 32 Gross, Kalinowski (1978) Fröbrich (1984) Potential Single folding Single folding Forces 1) Conservative 2) instant friction 1) Conservative, 2) instant friction 3) dynamical quadrupole deformations TMSF (2014) Double folding with DD M3Y & finite range exchange term 1) Conservative 2) retarding friction 3) thermal fluctuations L continuous continuous discrete Data ~20% ~20% ~1% Agreemen t Figures Figures χ² Density Fermi step Fermi step Microscopical Hartree-Fock approach

34 34 [7] H. de Vries et al., At. Dat. Nucl. Dat. Tab. 36 (1987) 495 Fig. 9 Parameters of the model: a a v np 2-1 ( ) ( ) K R : MeVzs τ : 0 1 zs aap : ( ) fm aat F nt ( ρ ) FA a a pp pt

35 36 RPT = RP + RT R= qr PT (30) (31) p q = pr PT (32) F q = FR PT (33) m m R A A = A A 2 P T q n PT P + T (34)

36 [6] MC, IG PRC 87 (2013) (no fluctuations) 8 TMSF DF2 M3Y K R = τ = MeV zs Fig. 7

37 9 Parameters of the model: K R = τ <1zs a a AP AT [ ] [ ] MeV zs 2 or MeV zs 3 : ( ) fm [2] D.H.E. Gross, H. Kalinowski, Phys. Rep. 45 (1978) 175 [3] P. Fröbrich, Phys. Rep. 116 (1984) 337

38 11 Fig. 12 [9] ANU data: PRC 52 (1995) 3151 σ exp > 200 mb τ = 0

39 Fig

40 Fig

41 Fig

42 Fig

43 12 Fig. 13 Dq t 0 Dq ( ) ( ) Φ = F s Γ t s ds (9)

44 13 O Sm ANU data: J.R. Leigh et al., PRC 52 (1995) 3151 ANU analysis: PRC 70 (2004) This work (2013) SBPM TMSF Double folding (M3Y Woods-Saxon potential DD6) a = 0.75 fm a = 0.68 fm WS χ = 2 ν 6.3 DF χ = 2 ν 3.4

45 λ = 2π p = MeV zs (35) (36) 37 O Zr pr 4 MeV zs fm 1 (37) λ =1.0 fm (38) RT = 4.9 fm (39) RT > λ (40)

46 The Need for Precise Experimental Above-Barrier Cross Sections: 15 Symmetric reactions: C O C O P+ P 32, 34, 36, 38 S+ S Cl + Cl , 48 40, 48 Ni + Ca Ni Ca Another above-barrier barrier cross sections: C+ Si, Ni, Ca, Y Si + Ni, Zr, Sm, Pb 31 P + spherical nuclei S+ Si, Ca, Ni, Zr, Sm 37 Cl + spherical nuclei , Ar + Sn, Sm, Pb, Bi

47 Double folding model 39 P Projectile U ( R) = U ( R) + U ( R) (41) PT n C R Target Fig. 4 T U ( ) nd R U ( ) ne R Direct part of the nuclear term: U R g dr dr ρ r F ρ v s ρ r (42) ( ) ( ) ( ) ( ) ( ) = nd P T PA P FA D TA T Exchange part of the nuclear term: UnE ( R) = g dr P drt ρpa ( rp ; rp + s ) F ρ v s ρ r ; r s exp ik s A (43) ( ) ( ) ( ) ( ) FA E TA T T rel red

48 Density dependence: 40 F { } ( ) C 1 exp( ) ρ = + α β ρ γ ρ FA F F F FA F FA ρ = ρ + + ρ ( r s 2) ( r s 2) FA PA P TA T (44) (45) P Fig. 20 T

49 Density dependence: 41 F { } ( ) C 1 exp( ) ρ = + α β ρ γ ρ (44) FA F F F FA F FA

50 42 Fig. 22 Fig. 21

51 ( ) ( ) ( ) p0 L = 2 mq Ec. m. UC q0 Un q0 Ucen q0, L (46) 43 ΔR 2 Un( R) = ln 1 + exp A0GK + A1 GK Δ R + A2GK Δ R agk 1/3 1/3 ( ) Δ R = R r A + A r0 GK = 1.30 fm GK P T a GK = 0.61 fm (48) A0 GK = 33 MeV A1 GK = 2 MeV A 2GK = 3 MeV (47) (49)

52 44 Fig. 23 Fig. 24

53 Fig. 25 Fig

54 46 2/3 ( ) 1 θ = E a A + a A DP( T ) 1 P( T ) 2 P( T ) (50) a 1 1 = MeV 1 a2 = MeV p E E E E U [20] A. V. Ignatyuk et al., Yad. Fiz. 21 (1975) q L D = DP + DT = cm 2 tot 2mq 2mqq 3 ( ) ( ) v ( s) = G [exp s r ] s r D Di vi vi i= 1 3 ( ) ( ) v ( s) = G [exp s r ] s r Ef Ef i vi vi i= 1 v Eδ( s) = GE δδ s ( ) (51) (52) (53)

55 SBPM DF2 M3Y TMSF DF2 M3Y 47 KR = zsmev 2 1 [6] J.O. Newton et al., PRC 64 (2001) [7] J.R. Leigh et al., PRC 52 (1995) 3151 [8] C.R. Morton et al., PRC 60 (1999) Fig. 27 Fig. 28

56 48 2 Lmax π σ = + 2mE R c. m. L= 0 ( 2L 1) T L (1) Fig. 29

57 SYSTEMATIC COMPARISON OF HEAVY-ION FUSION BARRIERS CALCULATED WITHIN THE FRAMEWORK OF THE DOUBLE FOLDING MODEL USING TWO VERSIONS OF NUCLEON-NUCLEON INTERACTION Gontchar I.I. 1, Chushnyakova M.V. 2,3 1 Omsk State Transport University, Omsk, Russia; 2 Omsk State Technical University, Omsk, Russia; 3 Tomsk Polytechnic University, Tomsk, Russia vigichar@hotmail.com It was shown in Refs. [1, 2] that the high energy parts of the heavy ion fusion excitation functions can be successfully reproduced within the framework of the double folding approach. The major ingredients of this approach are the nuclear densities and the effective nucleon-nucleon interaction [3]. In [1, 2] the wellknown M3Y NN forces [4, 5] were used. However sometimes in the literature the Migdal NN forces [6] are used to calculate the nucleus-nucleus potential [7, 8]. The question is to what extent the fusion barrier heights are different when calculating interaction potentials within the double folding approach with two options of the nucleon-nucleon interaction: the M3Y and the Migdal ones. In the present work we address this question performing systematic calculations of the fusion barrier height for zero angular momentum, U B0. In our calculations the nuclear densities came from the Hartree-Fock approach with the SKX coefficient set [9, 10]. The charge densities obtained within these calculations are shown to be in good agreement with the experimental data [11, 12]. The values of U B0 are calculated in the wide range of the value of the 1/3 1/3 parameter BZ ZPZT / AP AT : it varies from 10 MeV up to 150 MeV. Only spherical nuclei from 12 C up to 208 Pb are considered. Our calculations make it possible to draw definite conclusions about the applicability of the Migdal NN forces for describing the nucleus-nucleus collision. 1. I.I.Gontchar et al. // Phys. Rev. C V M.V.Chushnyakova et al. // Phys. Rev. C V Dao T.Khoa et al. // Phys. Rev. C V.56. P G.Bertsch et al. // Nucl. Phys. A V.284. P N.Anantaraman et al. // Nucl. Phys. A V.398. P A.B.Migdal. Theory of Finite Fermi Systems and Application to Atomic Nuclei. New York: Interscience Publishers P N.V.Antonenko et al. // Phys. Rev. C V.50. P V.I.Zagrebaev et al. // Phys. Elem. Part. At. Nucl V.38. P B.A.Brown // Phys. Rev. C V.58. P R.Bhattacharya // Nucl. Phys. A V.913. P H.de Vries et al. // At. Dat. Nucl. Dat. Tables V.36. P I.Angeli // At. Dat. Nucl. Dat. Tables V.87. P