Heavy-Ion Fusion Reactions around the Coulomb Barrier

Transcription

1 Heavy-Ion Fusion Reactions around the Coulomb Barrier Kouichi Hagino Tohoku University, Sendai, Japan cf. Experimental aspects of H.I. Fusion reactions: lectures by Prof. Mahananda Dasgupta (ANU)

2 3.11 earthquake Sendai after 1 month

3 Heavy-Ion Fusion Reactions around the Coulomb Barrier Kouichi Hagino Tohoku University, Sendai, Japan Fusion reactions and quantum tunneling Basics of the Coupled-channels method Concept of Fusion barrier distribution Quasi-elastic scattering and quantum reflection cf. Experimental aspects of H.I. Fusion reactions: lectures by Prof. Mahananda Dasgupta (ANU)

4 Fusion: compound nucleus formation courtesy: Felipe Canto

5 Inter-nucleus potential Two forces: 1. Coulomb force Long range, repulsive 2. Nuclear force Short range, attractive above barrier sub-barrier deep subbarrier Potential barrier due to the compensation between the two (Coulomb barrier)

6 Why subbarrier fusion? Two obvious reasons: discovering new elements (SHE by cold fusion reactions) nuclear astrophysics (fusion in stars)

7 Why subbarrier fusion? Two obvious reasons: discovering new elements (SHE) nuclear astrophysics (fusion in stars) Other reasons: reaction mechamism strong interplay between reaction and structure (channel coupling effects) cf. high E reactions: much simpler reaction mechanism many-particle tunneling cf. alpha decay: fixed energy tunneling in atomic collision: less variety of intrinsic motions

8 Basic of nuclear reactions Shape, interaction, and excitation structures of nuclei scattering expt. cf. Experiment by Rutherford (a scatt.) Notation Target nucleus b detector a X Y Projectile (beam) X(a,b)Y measures a particle intensity as a function of scattering angles 208 Pb( 16 O, 16 O) 208 Pb : 16 O+ 208 Pb elastic scattering 208 Pb( 16 O, 16 O ) 208 Pb : 16 O+ 208 Pb inelastic scattering 208 Pb( 17 O, 16 O) 209 Pb : 1 neutron transfer reaction

9 Scattering Amplitude =(incident wave) + (scattering wave)

10 Differential cross section dw The number of scattered particle through the solid angle of dw per unit time: (flux for the scatt. wave)

11 Scattering Amplitude partial wave decomposition Motion of Free particle: In the presence of a potential: Asymptotic form of wave function (scattering amplitude)

12 (note) Total incoming flux Total outgoing flux r r If only elastic scattering: (flux conservation) :phase shift

13 Optical potential and Absorption cross section Reaction processes Elastic scatt. Inelastic scatt. Transfer reaction Compound nucleus formation (fusion) Optical potential Loss of incident flux (absorption) (note) Gauss s law

14 Total incoming flux Total outgoing flux r r Net flux loss: Absorption cross section:

15 In the case of three-dimensional spherical potential: (reflection coeff.)

16 Overview of heavy-ion reactions Heavy-ion: Nuclei heavier than 4 He Inter-nucleus potential Two forces: 1. Coulomb force Long range, repulsive 2. Nuclear force Short range, attractive Potential barrier due to the compensation between these two (Coulomb barrier)

17 Double Folding Potential cf. Michigan 3 range Yukawa (M3Y) interaction (MeV) Phenomenological potential

18 Three important features of heavy-ion reactions 1. Coulomb interaction: important 2. Reduced mass: large (semi-) classical picture concept of trajectory 3. Strong absorption inside the Coul. barrier r touch 154 Sm 16 O r touch Strong absorption Automatic compound nucleus formation once touched (assumption of strong absorption)

19 Strong absorption the region of large overlap High level density (CN) Huge number of d.o.f. Relative energy is quickly lost and converted to internal energy :can access to the strong absorption :cannot access cassically Formation of hot CN (fusion reaction)

20 Partial decomposition of reaction cross section Taken from J.S. Lilley, Nuclear Physics

21 Classical Model for heavy-ion fusion reactions Strong absorption b (impact parameter) : can access to the strong absorption region classically

22 Classical fusion cross section is proportional to 1 / E pr b 2 1/V b Taken from J.S. Lilley, Nuclear Physics

23 Fusion reaction and Quantum Tunneling r touch 154 Sm 16 O r touch Automatic CN formation once touched (assumption of strong absorption) Strong absorption Probability of fusion = prob. to access to r touch Fusion takes place by quantum tunneling at low energies! Penetrability of barrier

24 Quantum Tunneling Phenomena V(x) V 0 -a a x V(x) x Tunnel probability:

25 For a parabolic barrier V b x

26 Energy derivative of penetrability (note) Classical limit

27 Potential Model: its success and failure Asymptotic boundary condition: Fusion cross section: Mean angular mom. of CN: r abs Strong absorption

28 Fusion cross section: Mean angular mom. of CN:

29 Wong s formula C.Y. Wong, Phys. Rev. Lett. 31 ( 73)766 i) Approximate the Coul. barrier by a parabola: ii) Approximate P l by P 0 : (assume l-independent Rb and curvature) iii) Replace the sum of l with an integral

30 (note) For (note)

31

32 Comparison between prediction of pot. model with expt. data Fusion cross sections calculated with a static energy independent potential 16 O+ 27 Al 14 N+ 12 C 40 Ar+ 144 Sm L.C. Vaz, J.M. Alexander, and G.R. Satchler, Phys. Rep. 69( 81)373 Works well for relatively light systems Underpredicts s fus for heavy systems at low energies

33 Potential model: Reproduces the data reasonably well for E > V b Underpredicts s fus for E < V b Data: J.R. Leigh et al., PRC52( 95)3151 cf. seminal work: R.G. Stokstad et al., PRL41( 78)465 PRC21( 80)2427 What is the origin? Inter-nuclear Potential is poorly parametrized? Other origins?

34 With a deeper nuclear potential (but still within a potential model)..

35 Potential Inversion (note) V b s fus V(r) r 1 r 2 r E Semi-classical app. Energy independent local single-ch.

36 Potential inversion s fus V(r) Semi-classical app. Energy independent local single-ch. 64 Ni + 64 Ni 64 Ni + 74 Ge Unphysical potentials Beautiful demonstration of C.C. effects A.B. Balantekin, S.E. Koonin, and J.W. Negele, PRC28( 83)1565

37 Potential inversion t(e) e.g., t(e) = 3 +/- 0.2 fm 64 Ni + 64 Ni 64 Ni + 74 Ge double valued potential (unphysical!!)

38 Fusion cross sections calculated with a static energy independent potential Potential model: Reproduces the data reasonably well for E > V b Underpredicts s fus for E < V b What is the origin? Inter-nuclear Potential is poorly parametrized? Other origins?

39 Target dependence of fusion cross section Strong target dependence at E < V b

40 Low-lying collective excitations in atomic nuclei Low-lying excited states in even-even nuclei are collective excitations, and strongly reflect the pairing correlation and shell strucuture Taken from R.F. Casten, Nuclear Structure from a Simple Perspective

41

42 Effect of collective excitation on s fus : rotational case Excitation spectra of 154 Sm cf. Rotational energy of a rigid body (Classical mechanics) (MeV) Sm is deformed Sm

43 Effect of collective excitation on s fus : rotational case Comparison of energy scales Tunneling motion: Rotational motion: 3.5 MeV (barrier curvature) The orientation angle of 154 Sm does not change much during fusion (note) Ground state (0 + state) when reaction starts 154 Sm 16 O Mixing of all orientations with an equal weight

44 Effect of collective excitation on s fus : rotational case The orientation angle of 154 Sm does not change much during fusion 154 Sm 16 O

45 154 Sm 16 O The barrier is lowered for =0 because an attraction works from large distances. The barrier increases for =p/2. because the rel. distance has to get small for the attraction to work Def. Effect: enhances s fus by a factor of 10 ~ 100 Fusion: interesting probe for nuclear structure

46 Two effects of channel couplings cf. 2-level model: Dasso, Landowne, and Winther, NPA405( 83)381 energy loss due to inelastic excitations no Coul-ex with Coul-ex dynamical modification of the Coulomb barrier large enhancement of fusion cross sections

47 One warning: Don t use this formula for light deformed nuclei, e.g., 28 Si 154 Sm 16 O (MeV) Si (MeV) 0

48 More quantal treatment: Coupled-Channels method Coupling between rel. and intrinsic motions Entrance channel Excited channel } excited states ground state

49 Schroedinger equation: or Coupled-channels equations

50 Coupled-channels equations boundary condition: equation for y k transition from f k to f k

51 Angular momentum coupling (MeV) I p = Total ang. mom.: Sm

52 Boundary condition (with ang. mom. coupling) Entrance channel Excited channel

53 Excitation structure of atomic nuclei Excite the target nucleus via collision with the projectile nucleus How does the targ. respond to the interaction with the proj.? Standard approach: analysis with the coupled-channels method Inelastic cross sections Elastic cross sections Fusion cross sections S-matrix S nli

54 How to perform coupled-channels calculations? 1. Modeling: selection of excited states to be included low-lying collective states only S. Raman et al., PRC43( 91)521

55 typical excitation spectrum: electron scattering data low-lying collective excitations GDR/GQR E GDR ~ 79A -1/3 MeV E GQR ~ 65A -1/3 MeV M. Sasao and Y. Torizuka, PRC15( 77)217 E x (MeV) low-lying non-collective excitations Giant Resonances: high E x, smooth mass number dependence adiabatic potential renormalization Low-lying collective excitations: barrier distributions, strong isotope dependence Non-collective excitations: either neglected completely or implicitly treated through an absorptive potential

56 2. Nature of collective states: vibration? or rotation? a) Vibrational coupling excitation operator: 0 +,2 +,

57 Vibrational excitations Bethe-Weizacker formula: Mass formula based on Liquid-Drop Model (A, Z) For a deformed shape, b a E e

58 In general E Harmonic oscillation a lm λ=2: Quadrupole vibration Movie: Dr. K. Arita (Nagoya Tech. U.)

59 In general E Harmonic oscillation a lm λ=3: Octupole vibration Movie: Dr. K. Arita (Nagoya Tech. U.)

60 Microscopic description Random phase approximation (RPA) Double phonon states Cd MeV MeV MeV MeV

61 2. Nature of collective states: vibration? or rotation? a) Vibrational coupling excitation operator: 0 +,2 +,

62 2. Nature of collective states: vibration? or rotation? b) Rotational coupling excitation operator:

63 Vibrational coupling Rotational coupling 0 +,2 +, cf. reorientation term

64 3. Coupling constants and coupling potentials Deformed Woods-Saxon model: excitation operator

65 Coupling Potential: Collective Model Vibrational case Rotational case Coordinate transformation to the body-fixed rame (for axial symmetry) In both cases (note) coordinate transformation to the rotating frame ( )

66 Deformed Woods-Saxon model (collective model) CCFULL K.H., N. Rowley, and A.T. Kruppa, Comp. Phys. Comm. 123( 99)143

67 CCFULL K.H., N. Rowley, and A.T. Kruppa, Comp. Phys. Comm. 123( 99)143 i) all order couplings Nuclear coupling: Coulomb coupling:

68 CCFULL K.H., N. Rowley, and A.T. Kruppa, Comp. Phys. Comm. 123( 99)143 i) all order couplings

69 CCFULL K.H., N. Rowley, and A.T. Kruppa, Comp. Phys. Comm. 123( 99)143 i) all order couplings K.H., N. Takigawa, M. Dasgupta, D.J. Hinde, and J.R. Leigh, PRC55( 97)276

70 CCFULL ii) isocentrifugal approximation J=l+I 2 + l=j-2, J, J l=j K.H., N. Rowley, and A.T. Kruppa, Comp. Phys. Comm. 123( 99)143 Truncation Dimension Iso-centrifugal approximation: λ: independent of excitations transform to the rotating frame Spin-less system

71 16 O Sm (2 + ) K.H. and N. Rowley, PRC69( 04)054610

72 CCFULL K.H., N. Rowley, and A.T. Kruppa, Comp. Phys. Comm. 123( 99)143 iii) incoming wave boundary condition (IWBC) (1) Complex potential r abs (2) IWBC limit of large W (strong absorption) (Incoming Wave Boundary Condition) strong absorption Only Real part of Potential More efficient at low energies cf. S l ~ 1 at low E

73 CCFULL K.H., N. Rowley, and A.T. Kruppa, Comp. Phys. Comm. 123( 99)143

74 ccfull.inp 16.,8.,144., ,-1,1.06,0 1.81,0.205,3,1 1.66,0.11,2,0 6.13,0.733,3,1 0,0., ,1.1, ,70.,1. 30,0.05 reaction system (A p =16, Z p =8, A t =144, Z t =62) r p, Ivibrot p, r t, Ivibrot t (inert projectile, and vib. for targ.)

75 ccfull.inp 16.,8.,144., ,-1,1.06,0 1.81,0.205,3,1 1.66,0.11,2,0 6.13,0.733,3,1 0,0., ,1.1, ,70.,1. 30,0.05 reaction system (A p =16, Z p =8, A t =144, Z t =62) r p, Ivibrot p, r t, Ivibrot t (inert projectile, and vib. for targ.) If Ivibrot t =0: O = O vib Ivibrot t =1: O = O rot Ivibrot t = -1: O = 0 (inert) similar for the projectile

76 ccfull.inp 16.,8.,144., ,-1,1.06,0 1.81,0.205,3,1 1.66,0.11,2,0 6.13,0.733,3,1 0,0., ,1.1, ,70.,1. 30,0.05 (A p =16, Z p =8, A t =144, Z t =62) (inert projectile, and vib. for targ.) properties of the targ. excitation E 1st = 1.81 MeV b = l = 3 N phonon = Sm coupling to 3 - vibrational state in the target with def. parameter b =

77 ccfull.inp 16.,8.,144., ,-1,1.06,0 1.81,0.205,3,1 1.66,0.11,2,0 6.13,0.733,3,1 (A p =16, Z p =8, A t =144, Z t =62) (inert projectile, and vib. for targ.) properties of the targ. excitation (note) if N phonon = 2: double phonon excitation 0,0., ,1.1, ,70.,1. E 1st = 1.81 MeV b = l = 3 N phonon = x Sm (3 - ) ,0.05

78 ccfull.inp 16.,8.,144., ,-1,1.06,0 1.81,0.205,3,1 1.66,0.11,2,0 6.13,0.733,3,1 (A p =16, Z p =8, A t =144, Z t =62) (inert projectile, and vib. for targ.) properties of the targ. excitation (note) if Ivibrott = 1 (rot. coup.) the input line would look like: 0.08,0.306,0.05,3 instead of 1.81,0.205,3,1 0,0., ,1.1, ,70.,1. 30,0.05 E 2+ b 2 b 4 N rot 3 excitated states (N rot =3) + g.s. 6x7x0.08/6 4x5x0.08/

79 ccfull.inp 16.,8.,144., ,-1,1.06,0 1.81,0.205,3,1 1.66,0.11,2,0 6.13,0.733,3,1 0,0., ,1.1, ,70.,1. 30,0.05 (A p =16, Z p =8, A t =144, Z t =62) (inert projectile, and vib. for targ.) properties of the targ. excitation same as the previous line, but the second mode of excitation in the target nucleus (vibrational coupling only) N phonon = 0 no second mode

80 ccfull.inp 16.,8.,144., ,-1,1.06,0 1.81,0.205,3,1 1.66,0.11,2,1 6.13,0.733,3,1 0,0., ,1.1, ,70.,1. 30,0.05 (A p =16, Z p =8, A t =144, Z t =62) (inert projectile, and vib. for targ.) properties of the targ. excitation second mode in the targ. (note) if N phonon = 1: the code will ask you while you run it whether your coupling scheme is (a) or (b) b 3 (a) b Sm b 3 (b) b Sm 3 - x

81 ccfull.inp 16.,8.,144., ,-1,1.06,0 1.81,0.205,3,1 1.66,0.11,2,1 6.13,0.733,3,1 0,0., ,1.1, ,70.,1. 30,0.05 (A p =16, Z p =8, A t =144, Z t =62) (inert projectile, and vib. for targ.) properties of the targ. excitation second mode in the targ. properties of the proj. excitation (similar as the third line) (will be skipped for an inert projectile)

82 ccfull.inp 16.,8.,144., ,-1,1.06,0 1.81,0.205,3,1 1.66,0.11,2,1 6.13,0.733,3,1 0,0.,0.3 (A p =16, Z p =8, A t =144, Z t =62) (inert projectile, and vib. for targ.) properties of the targ. excitation second mode in the targ. properties of the proj. excitation (similar as the third line) transfer coupling (g.s. to g.s.) 105.1,1.1, ,70.,1. (A p + A t ) Q tr = +3 MeV (A p + A t ) 30,0.05 * no transfer coup. for F = 0

83 ccfull.inp 16.,8.,144., ,-1,1.06,0 1.81,0.205,3,1 1.66,0.11,2,1 6.13,0.733,3,1 0,0., ,1.1, ,70.,1. 30,0.05 (A p =16, Z p =8, A t =144, Z t =62) (inert projectile, and vib. for targ.) properties of the targ. excitation second mode in the targ. properties of the proj. excitation (similar as the third line) transfer coupling (g.s. to g.s.) potential parameters V 0 = MeV, a = 0.75 fm R 0 = 1.1 * (A p 1/3 + A t 1/3 ) fm

84 ccfull.inp 16.,8.,144., ,-1,1.06,0 1.81,0.205,3,1 1.66,0.11,2,1 6.13,0.733,3,1 0,0., ,1.1, ,70.,1. 30,0.05 (A p =16, Z p =8, A t =144, Z t =62) (inert projectile, and vib. for targ.) properties of the targ. excitation second mode in the targ. properties of the proj. excitation (similar as the third line) transfer coupling (g.s. to g.s.) potential parameters E min, E max, DE (c.m. energies) R max, Dr

85 ccfull.inp 16.,8.,144., ,-1,1.06,0 1.81,0.205,3,1 1.66,0.11,2,1 6.13,0.733,3,1 0,0., ,1.1, ,70.,1. 30,0.05 OUTPUT 16O + 144Sm Fusion reaction Phonon Excitation in the targ.: beta_n= 0.205, beta_c= 0.205, r0= 1.06(fm), omega= 1.81(MeV), Lambda= 3, Nph= Potential parameters: V0= (MeV), r0= 1.10(fm), a= 0.75(fm),power= 1.00 Uncoupled barrier: Rb=10.82(fm), Vb= 61.25(MeV), Curv=4.25(MeV) Ecm (MeV) sigma (mb) <l> E In addition, cross.dat : fusion cross sections only

86 Coupled-channels equations and barrier distribution Calculate s fus by numerically solving the coupled-channels equations Let us consider a limiting case in order to understand (interpret) the numerical results e ni : very large e ni = 0 Adiabatic limit Sudden limit

87 Comparison of two time scales spring on a board kdl mg sin mg mg cos static case: mg sin = kdl Dl = mg sin / k

88 move very slowly? or move instantaneously?

89 Comparison of two time scales similar related example: spring on a moving board move very slowly? or move instantaneously? keep the original length (Dl =0) sudden limit always at the equilibrium length (Dl = mg sin / k ) adiabatic limit

90 fast reaction large fluctuation slow reaction + small fluctuation around the adiabatic path relative distance

91 Two limiting cases: (i) adiabatic limit much slower rel. motion than the intrinsic motion much larger energy scale for intrinsic motion than the typical energy scale for the rel. motion (Barrier curvature v.s. Intrinsic excitation energy)

92 c.f. Born-Oppenheimer approximation for hydrogen molecule r e - p R p 1. Consider first the electron motion for a fixed R 2. Minimize e n (R) with respect to R Or 2. Consider the proton motion in a potential e n (R)

93 Adiabatic Potential Renormalization 16 O(3 - ) Sm(3 - ) When e is large, where Fast intrinsic motion Adiabatic potential renormalization Giant Resonances, 16 O(3 - ) [6.31 MeV] K.H., N. Takigawa, M. Dasgupta, D.J. Hinde, J.R. Leigh, PRL79( 99)2014

94 typical excitation spectrum: electron scattering data low-lying collective excitations GDR/GQR E GDR ~ 79A -1/3 MeV E GQR ~ 65A -1/3 MeV M. Sasao and Y. Torizuka, PRC15( 77)217 E x (MeV) low-lying non-collective excitations Giant Resonances: high E x, smooth mass number dependence adiabatic potential renormalization Low-lying collective excitations: barrier distributions, strong isotope dependence Non-collective excitations: either neglected completely or implicitly treated through an absorptive potential

95 Two limiting cases: (ii) sudden limit 154 Sm 16 O Coupled-channels: diagonalize Slow intrinsic motion Barrier Distribution

96 Barrier distribution w P 0 B 1 B 2 B 3 B E B E B 1 B 2 B 3 E

97 Barrier distribution: understand the concept using a spin Hamiltonian Hamiltonian (example 1): For Spin-up For Spin-down x x

98 Wave function (general form) The spin direction does not change during tunneling:

99 Tunneling prob. is a weighted sum of tunnel prob. for two barriers

100 Tunnel prob. is enhanced at E < V b and hindered E > V b dp/de splits to two peaks barrier distribution The peak positions of dp/de correspond to each barrier height The height of each peak is proportional to the weight factor

101 simple 2-level model (Dasso, Landowne, and Winther, NPA405( 83)381) entrance channel excited channel

102 simple 2-level model (Dasso, Landowne, and Winther, NPA405( 83)381)

103 Sub-barrier Fusion and Barrier distribution method (Fusion barrier distribution) N. Rowley, G.R. Satchler, P.H. Stelson, PLB254( 91)25

104 N. Rowley, G.R. Satchler, P.H. Stelson, PLB254( 91)25 centered on E= V b

105 Barrier distribution measurements Fusion barrier distribution Needs high precision data in order for the 2 nd derivative to be meaningful (early 90 s)

106 Experimental Barrier Distribution Requires high precision data T 154 Sm 16 O M. Dasgupta et al., Annu. Rev. Nucl. Part. Sci. 48( 98)401

107 Investigate nuclear shape through barrier distribution Nuclear shapes

108 By taking the barrier distribution, one can very clearly see the difference due to b 4! Fusion as a quantum tunneling microscope for nuclei

109 Advantage of fusion barrier distribution Fusion Cross sections Very strong exponential energy dependence Difficult to see differences due to details of nuclear structure Plot cross sections in a different way: Fusion barrier distribution N. Rowley, G.R. Satchler, P.H. Stelson, PLB254( 91)25 Function which is sensitive to details of nuclear structure

110 Example for spherical vibrational system 16 O Sm Anharmonicity of octupole vibration Sm 0 + Quadrupole moment: K.Hagino, N. Takigawa, and S. Kuyucak, PRL79( 97)2943

111 Barrier distribution 16 O Sm 1.8 MeV Sm 0 + K.Hagino, N. Takigawa, and S. Kuyucak, PRL79( 97)2943

112 Coupling to excited states distribution of potential barrier multi-dimensional potential surface single barrier a collection of many barriers relative distance r x (intrinsic coordinate)

113 Representations of fusion cross sections i) s fus vs 1/E (~70 s) Classical fusion cross section is proportional to 1 / E : pr b 2 1/V b Taken from J.S. Lilley, Nuclear Physics

114 ii) barrier distribution (~90 s) M. Dasgupta et al., Annu. Rev. Nucl. Part. Sci. 48( 98)401

115 iii) logarithmic derivative (~00 s) cf. R. Vandenbosch, Ann. Rev. Nucl. Part. Sci. 42( 92)447 M. Dasgupta et al., PRL99( 07)

116 deep subbarrier hindrance of fusion cross sections C.L. Jiang et al., PRL89( 02)052701; PRL93( 04)012701

117 Systematics of the touching point energy and deep subbarrier hindrance mechanism of deep subbarrier hindrance: not yet been fully understood how to model the dynamics after touching? T. Ichikawa, K.H., A. Iwamoto, PRC75( 07) &

118 Quantum reflection and quasi-elastic scattering V b x In quantum mechanics, reflection occurs even at E > V b Quantum Reflection Reflection prob. carries the same information as penetrability, and barrier distribution can be defined in terms of reflection prob.

119 Quasi-Elastic Scattering A sum of all the reaction processes other than fusion (elastic + inelastic + transfer + ) Fusion Quasi-elastic Detect all the particles which reflect at the barrier and hit the detector In case of a def. target Related to reflection Complementary to fusion T 154 Sm 16 O

120 Subbarrier enhancement of fusion cross sections 16 O Sm Quasi-elastic scattering (elastic + inelastic) 16 O Sm 154 Sm 16 O

121 Quasi-elastic barrier distribution T 154 Sm 16 O Quasi-elastic barrier distribution: H. Timmers et al., NPA584( 95)190 (note) Classical elastic cross section in the limit of strong Coulomb field:

122 Quasi-elastic test function Classical elastic cross section (in the limit of a strong Coulomb): Nuclear effects Semi-classical perturbation theory S. Landowne and H.H. Wolter, NPA351( 81)171 K.H. and N. Rowley, PRC69( 04)054610

123 Quasi-elastic test function The peak position slightly deviates from V b Low energy tail Integral over E: unity Relatively narrow width Close analog to fusion b.d.

124

125 Scaling property 16 O Sm Expt.: impossible to perform at Relation among different? Effective energy:

126 16 O Sm

127 Comparison of D fus with D qel Fusion Quasi-elastic K.H. and N. Rowley, PRC69( 04) H. Timmers et al., NPA584( 95)190 A gross feature is similar to each other

128 Experimental barrier distribution with QEL scattering H. Timmers et al., NPA584( 95) Zn Pb 70 Zn : E 2 = MeV, 2 phonon, 208 Pb: E 3 = MeV, 3 phonon Muhammad Zamrun F., K. H., S. Mitsuoka, and H. Ikezoe, PRC77( 08) Experimental Data: S. Mitsuoka et al., PRL99( 07)182701

129 Experimental advantages for D qel less accuracy is required in the data (1 st vs. 2 nd derivative) much easier to be measured Qel: a sum of everything a very simple charged-particle detector Fusion: requires a specialized recoil separator to separate ER from the incident beam ER + fission for heavy systems several effective energies can be measured at a single-beam energy relation between a scattering angle and an impact parameter

130 Deep subbarrier fusion and diffuseness anomaly Scattering processes: Double folding pot. Woods-Saxon (a ~ 0.63 fm) successful A. Mukherjee, D.J. Hinde, M. Dasgupta, K.H., et al., PRC75( 07) Fusion process: not successful a ~ 1.0 fm required (if WS)

131 scattering fusion How reliable is the DFM/WS? What is an optimum potential? deduction of fusion barrier from exp. data? (model independent analysis?)

132 Quasi-elastic scattering at deep subbarrier energies? K.H., T. Takehi, A.B. Balantekin, and N. Takigawa, PRC71( 05) K. Washiyama, K.H., M. Dasgupta, PRC73( 06) QEL at deep subbarrier energies: sensitive only to the surface region 16 O Sm

133 Summary Heavy-Ion Fusion Reactions around the Coulomb Barrier Fusion and quantum tunneling Fusion takes place by tunneling Basics of the Coupled-channels method Collective excitations during fusion Concept of Fusion barrier distribution Sensitive to nuclear structure Quasi-elastic scattering and quantum reflection Complementary to fusion Computer program: CCFULL

134 References Nuclear Reaction in general G.R. Satchler, Direct Nuclear Reactions G.R. Satchler, Introduction to Nuclear Reaction R.A. Broglia and A. Winther, Heavy-Ion Reactions Treatise on Heavy-Ion Science, vol. 1-7 D.M. Brink, Semi-classical method in nucleus-nucleus collisions P. Frobrich and R. Lipperheide, Theory of Nuclear Reactions Heavy-ion Fusion Reactions M. Dasgupta et al., Annu. Rev. Nucl. Part. Sci. 48( 98) 401 A.B. Balantekin and N. Takigawa, Rev. Mod. Phys. 70( 98) 77 Proc. of Fusion03, Prog. Theo. Phys. Suppl. 154( 04) Proc. of Fusion97, J. of Phys. G 23 ( 97) Proc. of Fusion06, AIP, in press.

135

136 Hamiltonian (example 3): more general cases x dependent E dependent K.H., N. Takigawa, A.B. Balantekin, PRC56( 97)2104 (note) Adiabatic limit: