Nuclear Reaction Mechanisms in Heavy Ion Collisions (Lecture II) 1 Joint Institute for Nuclear Research, Dubna, Russia NASIROV Avazbek * 2014 International SCHOOL, May VECC Kolkata 13 May 2014 * Institute of Nuclear Physics, Tashkent, Uzbekistan

My schedule of talks ) Nuclear Reaction Mechanisms in Heavy Ion Collisions ) Entrance channel effect in formation of the reaction products ) Mass and angular distribution of the capture reaction products ) Dynamics of complete fusion and quasifission reactions.

Example of mixing products formed in the different reaction channels

Eloss < 5 MeV 5 MeV < Eloss 5 <Eloss < 40 MeV 5 <Eloss < 150 MeV

Interaction forces between colliding nuclei R A 1, Z 1 A2, Z2A2, Z2 R 1, R 2 << R R 1 + R 2 ~ R Z1Z1 Z2Z2 Nuclear forces are included into interaction between nucleus. Nucleons attract each other via the strong nuclear force ( range ~ 1 fm)

Basic quantities for colliding nuclei P b E c.m., A p, A T, Z P, Z T L=[b x P] R E c.m. =E lab A T /(A P +A T ) A P A T 0.17 fm -3

is a scattering angle

The differential cross section observed for electrons of 153 MeV scattered from a gold target. It is seen that, for the angles studied, the intensity of the scattering is at least an order of magnitude weaker than for a point charge of Z = 79. The angular distribution exhibits mild oscillations characteristic of scattering by a system with a rather well-defined radius The experimental data and the theoretical analysis are taken from B.Hahn, et al. Phys. Rev. 101, 1131 (1956); D. R. Yennie, et al, Phys. Rev. 95,500 (1954); R. Herman and R. Hofstadter, High Energy Electron Scattering Tables, Stanford Univ. Press, Stanford, California, Validity of the mean-field approximation based on the analysis of the experimental data

Main characteristics of reaction products which is observed. G. Guarino et al.,

10 1. Nucleons MassSpinCharge Proton MeV/c 2 1/2+1e Neutron MeV/c 2 1/20 size: ~1 fm 2. Nuclei a bunch of nucleons bound together create a potential for an additional : nucleons attract each other via the strong force ( range ~ 1 fm) neutron proton (or any other charged particle) V r R V r R Coulomb Barrier V c Potential … … Nucleons in a Box: Discrete energy levels in nucleus R ~ 1.3 x A 1/3 fm 1 a.m.u MeV/c 2

Shell model: (single nucleon energy levels) Magic numbers are not evenly spaced shell gaps more bound than average less bound than average need to add shell correction term S(Z,N)

Theoretical methods to describe multinucleon transfer reactions at low energies

15 Hamiltonian for calculation of the transport coefficients The macroscopic motion of nucleus and microscopic motion of nucleons must be calculated simultaneously.

16 Evolution operator for the macroscopic and microscopic degrees of freedom is defined by solution of the Schroedinger equation with initial condition U(t i,t i )=1 : U’ is defined by coupling term between collective and microscopic variables A.Messia, “Quantum mechanics” Vol.1

17 Equations of motion used to find capture of projectile by target

Nucleus-nucleus interaction potential 18

Constants of the effective nucleon-nucleon interaction (Migdal forces) 19

20 Master equations for the nucleon occupation numbers and Equation of motion for the relative distance (5) (6) (7) (8) (9). G.G. Adamian, et al. Phys. Rev. C53, (1996) p

Solutions for the density matrixes where

Time evolution of the occupation numbers of protons and neutrons in interacting nuclei.

Calculation of the physical quantities characterizing multinucleon transfer reactions

24 Friction coefficients  j and  k are single particle energies of components of dinuclear system; is a width of the excited single-particle states due to residual interaction between nucleons. G.G. Adamian, et al. Phys. Rev. C56 No.2, (1997) p

Width of the single-particle excited states A. Nasirov et al, Nucl. Phys. A 759 (2005) 342–369 D. Pines, P. Noziéres, Theory of Quantum Liquids, Benjamin, New York, 1966

26 The change of nuclear shape The change of the nuclear shape was taken into account by solving of equations of motion for the quadrupole (2+) and octupole (3-) collective excitations in nucleus i (i=1,2): where ,  and D are the frequency, damping and mass coefficients for the surface vibration multipolarity, respectively; R0 is radius of the spherical nucleus. The values of  and reduced electric- multipole transition rate BE are obtained from the Tables in G. Audi, A.H. Wapstra, Nucl. Phys. A 595, 509 (1995). The damping coefficient  is calculated from the estimation of the coupling term between surface vibrations and nucleon motion in nuclei

Nucleus-Nucleus Collisions at low energies

Difference between deep-inelastic collisions and capture events ΔE L=0 It is important relations between ΔE and difference E c.m. –V min as well as between E c.m. –V min and depth of the potential well B qf. B qf. V min

29 Comparison of the friction coefficients calculated by the different methods -- Gross-Kalinovski Solid line – semi-microscopic method Long dashed __ Short dashed.- - Dotted …. Temperature= 2 МэВ Temperature = 1 МэВ Temperature = 0.5 МэВ Linear response theory by Yamaji Phys. Rev. C56 No.2, (1997) p

30 The friction coefficient depends on the relative distance between centers of nuclei and increases by temperature of nuclei. Dotted curve Solid curve …... ___ Incoming path Outgoing path

Hindrance to formation of compound nucleus

P T P’ T’ P” T” Deep – inelastic collisions ( I ) ( I I ) ( I I I ) ( V ) Heavy ion collisions Dinuclear system Quasifission Compound nucleus Complete fusion Capture Evaporation residues Fusion-fission F1F1 F2F2 Cooling Mononucleus ( I V ) F1F1 F2F2 Fast fission  ER (E Lab,L) =  cap (E Lab,L) P CN (E Lab,L) W surv (E Lab,L  {  } E c.m. =E lab

G.G. Adamian, G. Giardina, A.K. Nasirov, in Cont. of "XIV Int. Workshop on Nuclear Fission" Physics, Obninsk, 1998, Russia, 2000 Yu.Ts Oganessian et al,Phys.Rev.C (2004) W.Q. Shen et al Phys.Rev. C (1987) GSI experiment Capture and fusion cross sections for the 48 Ca+ 238 U reaction.

34 Strong decreasing fusion probability for synthesis superheavy elements

35

Fission barriers calculated by macroscopic-microscopic model: M. Kowal,P.Jachimowicz,and A. Sobiczewski, Phys. Rev. C 82, (2010)

Z= Ca+ 232 Th→ * * → n Z=110; N=167 No synthesis !

Fission barriers calculated by macroscopic-microscopic model: M. Kowal,P.Jachimowicz,and A. Sobiczewski, Phys. Rev. C 82, (2010) Z= Ca+ 238 U→ * * → n Z=112; N=171 Small cross section was observed in Dubna !

Fission barriers calculated by macroscopic-microscopic model: M. Kowal,P.Jachimowicz,and A. Sobiczewski, Phys. Rev. C 82, (2010) Z= Ni+ 208 Pb → * * → n Z=110; N=161 Large cross section ! Darmstadtium was obtained in Germany

40 Theoretical approaches to describe and to predict the cross sections and favorable beam energies leading to new superheavy element Z= Zhao-Qing Feng, Gen-Ming Jin, Jun-Qing Li, and Werner Scheid, Phys. Rev. C 76, (2007). 2. V. I. Zagrebaev and W. Greiner, Phys. Rev. C 78, (2008). 3. G.G. Adamian N.V. Antonenko, and W. Scheid, Eur. Phys. J. A 41, 235 (2009). 4. A. K. Nasirov, G. Giardina, S.Hofmann, et al. Phys. Rev. C 79, (2009). 50 Ti+ 249 Cf ⟶

Potential energy surface of dinuclear system a- entrance channel; b-fusion channel; c and d are quasifission channels U dr (A, Z,, ß 1, ß 2 ) = B 1 + B 2 + V (A, Z, ß 1 ; ß 2 ; R) - B C N - V C N (L ) G. Giardina, S. Hofmann, A.I. Muminov, and A.K. Nasirov, Eur. Phys. J. A 8, 205–216 (2000)

42 Potential energy surface of dinuclear system formed in 48 Ca+ 248 Cm reaction Comparison of the capture, fusion, quasifission and fast fission cross sections in 48 Ca+ 248 Cm reaction

43 Driving potential U driving ( c ) for reactions 40 Ar+ 172 Hf, 86 Kr+ 130 Xe, 124 Sn+ 92 Zr leading to formation of compound nucleus 216 Th : U driving =B 1 +B 2 -B (1+2) +V( R ) Due to peculiarities of shell structure B fus (Kr) > Bfus (Kr) and, consequently,  fus (Kr+Xe) <  fus (Zr+Sn)

The change of driving potential by increase of the mass and charge of compound nucleus. 44

Theoretical calculation of evaporation residue cross section (synthesis of superheavy element). Is fusion probability which calculated by diffusion-dissipative method, i.e. by solving Langeven equation or G.G. Adamian N.V. Antonenko, and W. Scheid, Eur. Phys. J. A 41, 235 (2009);. A. K. Nasirov, G. Giardina, S.Hofmann, et al. Phys. Rev. C 79, (2009).. is capture probability., which calculated in different theoretical models by different way.

Calculation of the competition between complete fusion and quasifission: P cn (E DNS,L) 46 Fazio G. et al, Modern Phys. Lett. A 20 (2005) p.391

Map of superheavy elements region 47 From paper Yuri Oganessian, Pure Appl. Chem., Vol. 78, No. 5, pp. 889–904,

Importance of the shell effects in compound nuclei in formation of evaporation residues-superheavy elements (SHE) 48 - Cold fusion A Z X +208 Pb, 209 Bi A Z X=Cr, Fe, Ni, Zn Hot fusion reactions 48 Ca+U, Pu, Cm, Cf

Results of calculation and comparison of them with the experimental data for the “cold” 64 Ni+ 208 Pb and 70 Zn+ 208 Pb reactions.. * RIKEN * GSI G.Giardina, et al. Eur. Phys. J. A 8, 205–216 (2000) S. Hofmann, Rep. Progr. Phys. 61, 639 (1998);

50

51 Angular momentum distribution for the complete fusion σ fus (L) (E lab ) as a function of momentum and and beam energy for reactions leading to formation of 216 Th. G. Fazio, et al., Journal of the Physical Society of Japan 388, 2509 (2003).

52 Driving potential U driving for reactions 40 Ar+ 172 Hf, 86 Kr+ 130 Xe, 124 Sn+ 92 Zr leading to formation of compound nucleus 216 Th : U driving =B 1 +B 2 -B (1+2) +V( R ) В 1 and В 2 binding energies of nuclei 1 and 2 are taken from the tables (top panel), if В 1 and В 2 are calculated by the liquid –drop model (middle panel) Quasifission barriers (bottom panel) G. Fazio, et al., Journal of the Physical Society of Japan 388, 2509 (2003).

53 Dependence of the intrinsic fusion barrier on angular momentum G. Fazio, et al Eur. Phys. J. A 19, 89 ・ 04 (2004) Udr = B 1 + B 2 - (B CN + V CN (L )) + V (A, Z, ß 1, α 1 ; ß 2, α 2 ; R,L)

F Fusion hindrance increases by increasing the orbital angular momentum. 54 Dependence of the driving potential and quasifission barrier on the angular momentum of dinuclear system formed in reactions leading to formation of compound nucleus 216 Th.

Dependence of the fission barrier on the excitation energy and angular momentum of compound nucleus. G.Giardina, et al. Eur. Phys. J. A 8, 205–216 (2000)

The role of the entrance channel was reduced by authors of paper to the difference in the critical values of angular momentum of the excited and rotating compound nucleus

Explanation by authors of experiments

The difference in the angular momentum distribution is caused by quasifission. The critical values of L are very close

We have explained the observed difference in the evaporation residues cross sections normalized to the fusion cross section as the result of counting the quasifission fragments as fusion-fission fragments by the authors of experiments.

Our interpretation of the observed difference in ratio R

What we know about quasifission fragments? The mass distribution its fragments has a maximum usually near magic numbers Z=20, 28, 50, 82 and N=20, 28, 50, 82; Total kinetic energy distribution is very close to Viola systematics as for fusion-fission: TKE=Z 1 Z 2 e 2 /D(A 1,A 2 ); Angular distribution of fragments has more large anisotropy in comparison with that of fusion-fission. angular distribution of quasifission fragments is mainly anisotropic but it may be isotropic and angular distributionof fusion-fission fragments may be isotropic in dependence on the reaction dynamics. We would like to stress that angular distribution of quasifission fragments is mainly anisotropic but it may be isotropic and angular distribution of fusion-fission fragments may be isotropic in dependence on the reaction dynamics.

Mixing of the distribution of fragment masses versus total kinetic energy W.Q. Shen et al (GSI) Phys.Rev.C36, 115 (1987)

Mixing of distribution of fragment masses versus center-of-mass angle W.Q. Shen et al (GSI) Phys.Rev.C36, 115 (1987)

Main assumptions of the dinuclear system model to study dissipative and quasifission processes.

Z P,M P Z T,M T i R There are two interacting nuclei – projectile and target which have Z P and Z T protons, and N P and N T neutrons, respectively. The protons and neutrons are placed on the corresponding single-particle states created by the mean-fields U P and U T. The quantum numbers and energies of these single-particle states are found by solving the Schrödinger equation with the Hamiltonians for both nuclei

Calculation of the physical quantities characterizing multinucleon transfer reactions

G.G. Adamian, R.V. Jolos, A.K. Nasirov, “ Partition of excitation energy between reaction products in heavy ion collisions”, ZEITSCHRIFT FOR PHYSIK A 347, (1994) The role of the particle-hole excitations and the nucleon exchange is considered. The ratio of the projectile excitation energy to the total excitation energy for the reactions 238 U(1468 MeV)+ 124 Sn, 238 U(1398 MeV)+ 124 Sn 56 Fe(505 MeV)+ 165 Ho, 74 Ge (629 MeV)+ 165 Ho and 68 Ni(880 MeV)+ 197 Au is calculated. The results of calculations are in good agreement with the experimental data.

Comparison of the calculated results for mean values of the charge and mass numbers in 56 Fe+ 165 Ho and 74 Ge+ 165 Ho reactions.

Ratio of the projectile-like product excitation energy to the total excitation energy

Thermal equilibrium is not reached in deep inelastic collisions… T P = T T TP T P > T T In case of the thermal equilibrium we have A reason of the non-equilibrium distribution is not only short time of the interaction time. The main reason is the nuclei have shell structure and it is changed by the change of proton and neutron numbers.

The well known experiments of fission where fission products are measured in coincidence with the neutron emission showed that from the heavy fragment neutrons emitted less than from light fragment. This fact confirms decisive role of shell structure in excitation distribution between reaction products. Thermal equilibrium is not reached in fusion- fission reactions too ?! E.M. Kozulin et al. Int. Conf. Fusion-06, Venezia

A new phenomenon has been observed in heavy ion reactions and has been termed “multinucleon transfer”, “quasifission”, “relaxation phenomena”, and “deep inelastic scattering”. These reactions are characterized by energy equilibration without mass equilibration, resulting in (i) two fragments with masses close to the target and projectile masses, (ii) fragment kinetic energies close to the calculated Coulomb repulsion of two normal fission fragments, and (iii) angular distributions distinct from those for complete fusion-fission. For ions with Z ~ 18, these new reactions were

Mixing of the distribution of the fragment masses versus total kinetic energy W.Q. Shen et al (GSI) Phys.Rev.C36, 115 (1987) 77

Mixing of distribution of fragment masses versus center-of-mass angle W.Q. Shen et al (GSI) Phys.Rev.C36, 115 (1987) 78

Correlation between total energy loss and variance of the proton distribution in deep-inelastic coollisions 79

Nucleus-nucleus interaction potential

Difference between deep-inelastic collisions and capture events ΔE L=0 It is important relations between ΔE and difference E c.m. –V min as well as between E c.m. –V min and depth of the potential well B qf. B qf. V min

Cross sections are found by collision dynamics of projectile and target-nucleus  cap (E lab,L;  1,  2 )= (2L +1) T(E lab, L;  1,  2 ) L dyn and L min are determined by dynamics of collision and calculated by solution of equations of motion for the collision trajectory:

Classical equations of the radial and tangential motions

P T P’ T’ P” T” Deep – inelastic collisions ( I ) ( I I ) ( I I I ) ( V ) Heavy ion collisions Dinuclear system Quasifission Compound nucleus Complete fusion Capture Evaporation residues Fusion-fission F1F1 F2F2 Cooling Mononucleus ( I V ) F1F1 F2F2 Fast fission  ER (E Lab,L) =  cap (E Lab,L) P CN (E Lab,L) W surv (E Lab,L  {  } E c.m. =E lab

G.G. Adamian, G. Giardina, A.K. Nasirov, in Cont. of "XIV Int. Workshop on Nuclear Fission" Physics, Obninsk, 1998, Russia, 2000 Yu.Ts Oganessian et al,Phys.Rev.C (2004) W.Q. Shen et al Phys.Rev. C (1987) GSI experiment Capture and fusion cross sections for the 48 Ca+ 238 U reaction.

Results of calculation and comparison of them with the experimental data for the “cold” 64 Ni+ 208 Pb and 70 Zn+ 208 Pb reactions.. * RIKEN * GSI G.Giardina, et al. Eur. Phys. J. A 8, 205–216 (2000) S. Hofmann, Rep. Progr. Phys. 61, 639 (1998);

87

88 Fission products

Energy-mass distribution of the reaction products in the 238 U (7.5 MeV/u) + 35 Cl reaction

About ambiguity of separation of fusion-fission (ff) and quasifission (qf) products at the analysis of experimental data.

Fission fragment angular distribution for the 32 S+ 184 W reaction. Incident energies are shown in the figure. The experimental data are shown with the fitting curve, which is used to determine the anisotropy A exp of the fragment angular distribution and mean square values of angular momentum from these events. H. Q. Zhang et al, Phys. Rev. C 81, (2010)

Interpretation of the experimental data presented as fusion- fission data in the 32 S+ 184 W reaction

Dependence of competition between complete fusion and quasifission on energy and orbital angular momentum allows us to determine the angular momentum distribution of dinuclear system and compound nucleus.

Dependence of competition between complete fusion and quasifission on energy and orbital angular momentum ? !

96 Angular momentum distribution of compound nuclei formed in collisions with different orientations of the target 154 Sm at different values of the beam energy. G. Fazio et al. Jour. Phys. Soc. of Japan., Vol. 77, No. 12, No. 12, December, 2008,

Interpretation of the measured capture and fusion excitation functions by description of evaporation residue cross sections.

Characteristic Properties of Deeply Inelastic Collisions and Quasifission reactions

99 Non-equilibrium processes in heavy ion collisions At А 1 +А 2 → А 1 ’ +А 2 ’ usually Е 1 * : Е 2 * ≠ А 1 ’ : А 2 ’ (even at fission!). At thermodynamic equilibrium must be Т 1 =Т 2 → Е 1 * : Е 2 * = А 1 ’ : А 2 ’ i=P,T R.V. Jolos, Eur. Phys. Jour. A7, 2000, p

Both fission products are registered in coincidence.

Yield neutrons and gamma quanta in coincidence with the fission fragments

G.G. Adamian, R.V. Jolos, A.K. Nasirov, “ Partition of excitation energy between reaction products in heavy ion collisions”, ZEITSCHRIFT FOR PHYSIK A 347, (1994) The role of the particle-hole excitations and the nucleon exchange is considered. The ratio of the projectile excitation energy to the total excitation energy for the reactions 238 U(1468 MeV)+ 124 Sn, 238 U(1398 MeV)+ 124 Sn 56 Fe(505 MeV)+ 165 Ho, 74 Ge (629 MeV)+ 165 Ho and 68 Ni(880 MeV)+ 197 Au is calculated. The results of calculations are in good agreement with the experimental data.