Joint Institute for Nuclear Research Dubna Russia Nuclear
Joint Institute for Nuclear Research, Dubna, Russia Nuclear Reaction Mechanisms in Heavy Ion Collisions (Lecture II) NASIROV Avazbek* 2014 International SCHOOL, May 13 -24 VECC Kolkata 13 May 2014 *Institute of Nuclear Physics, Tashkent, Uzbekistan 1
My schedule of talks 13. 05. 2014 16. 30 -17. 30 1) Nuclear Reaction Mechanisms in Heavy Ion Collisions. 14. 05. 2014 16. 30 -17. 30 2) Entrance channel effect in formation of the reaction products. 15. 05. 2014 16. 30 -17. 30 3) Mass and angular distribution of the capture reaction products. 16. 05. 2014 15. 00 -16. 00 4) Dynamics of complete fusion and quasifission reactions.
Example of mixing products formed in the different reaction channels
Eloss < 5 Me. V < Eloss 5 <Eloss < 40 Me. V 5 <Eloss < 150 Me. V
Interaction forces between colliding nuclei • A 2, Z 2 A 1, Z 1 R R 1, R 2 << R R 1+ R 2 ~ R Z 1 Z 2 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 A P P A T b R Ec. m. =Elab AT /(AP +AT) 0. 17 fm-3 Ec. m. , Ap, AT, ZP, ZT L=[b x P] 0. 17 fm-3
is a scattering angle
Validity of the mean-field approximation based on the analysis of the experimental data The differential cross section observed for electrons of 153 Me. V 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 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, 1960.
Main characteristics of reaction products which is observed. G. Guarino et al. ,
1. Nucleons Mass Spin Charge Proton 938. 272 Me. V/c 2 1/2 +1 e Neutron 939. 565 Me. V/c 2 1/2 0 size: ~1 fm 1 a. m. u. -931. 06 Me. V/c 2 2. Nuclei nucleons attract each other via the strong force ( range ~ 1 fm) a bunch of nucleons bound together create a potential for an additional : neutron proton (or any other charged particle) V V R … Coulomb Barrier Vc Potential R ~ 1. 3 x A 1/3 fm r … R r Nucleons in a Box: Discrete energy levels in nucleus 10
Shell model: (single nucleon energy levels) are not evenly spaced 184 shell gaps 126 less bound than average 82 more bound than average 50 28 20 need to add shell correction term S(Z, N) 8 2 Magic numbers
Theoretical methods to describe multinucleon transfer reactions at low energies
Hamiltonian for calculation of the transport coefficients The macroscopic motion of nucleus and microscopic motion of nucleons must be calculated simultaneously. 15
Evolution operator for the macroscopic and microscopic degrees of freedom is defined by solution of the Schroedinger equation with initial condition U(ti, ti)=1 : U’ is defined by coupling term between collective and microscopic variables A. Messia, “Quantum mechanics” Vol. 1 16
Equations of motion used to find capture of projectile by target 17
Nucleus-nucleus interaction potential 18
Constants of the effective nucleon-nucleon interaction (Migdal forces) 19
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. C 53, (1996) p. 871 -879 20
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
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. C 56 No. 2, (1997) p. 373 -380 24
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
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; R 0 is radius of the spherical nucleus. The values of and reduced electricmultipole 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 26
Nucleus-Nucleus Collisions at low energies
Difference between deep-inelastic collisions and capture events • ΔE Vmin Bqf. L=0 It is important relations between ΔE and difference Ec. m. –Vmin as well as between Ec. m. –Vmin and depth of the potential well Bqf.
Comparison of the friction coefficients calculated by the different methods -- Gross-Kalinovski Solid line – semi-microscopic method Linear response theory by Yamaji Long dashed __ Temperature= 2 Мэ. В Short dashed. - - Temperature = 1 Мэ. В Dotted …. Temperature = 0. 5 Мэ. В Phys. Rev. C 56 No. 2, (1997) p. 373 -380 29
The friction coefficient depends on the relative distance between centers of nuclei and increases by temperature of nuclei. …. . . Dotted curve Incoming path Solid curve ___ Outgoing path 30
Hindrance to formation of compound nucleus
P P’ T Heavy ion collisions T’ Cap ture Deep – inelastic collisions ss Quasifi ( I ) ion ( I I ) P” T” Dinuclear system F 1 F 2 ( I I I ) sion Fast fis Mononucleus Ec. m. =Elab F 1 Co fus mpl ion ete F 2 ( I V) ( V ) g in ol Co Compound nucleus sion s i f n o i Fus ER (ELab , L) = cap(ELab , L) PCN(ELab , L) Wsurv (ELab , L { } Evaporation residues
Capture and fusion cross sections for the 48 Ca+238 U reaction. W. Q. Shen et al Phys. Rev. C 36. 115 (1987) GSI experiment Yu. Ts Oganessian et al, Phys. Rev. C 70 064609 (2004) G. G. Adamian, G. Giardina, A. K. Nasirov, in Cont. of "XIV Int. Workshop on Nuclear Fission" Physics, Obninsk, 1998, Russia, 2000
Strong decreasing fusion probability for synthesis superheavy elements 34
35
Fission barriers calculated by macroscopic-microscopic model: M. Kowal, P. Jachimowicz, and A. Sobiczewski, Phys. Rev. C 82, 014303 (2010)
Fission barriers calculated by macroscopic-microscopic model: M. Kowal, P. Jachimowicz, and A. Sobiczewski, Phys. Rev. C 82, 014303 (2010) 48 Ca+232 Th→ 280110 * Z=120 280110* → 277 110+3 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, 014303 (2010) 48 Ca+ 238 U→ 286112 * Z=120 286112* → 283 112+3 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, 014303 (2010) 64 Ni+ 208 Pb → 272110 * Z=120 272110* → 271 110+1 n Z=110; N=161 Large cross section ! Darmstadtium was obtained in Germany
Theoretical approaches to describe and to predict the cross sections and favorable beam energies leading to new superheavy element Z=120. 50 Ti+249 Cf ⟶ 299120 1. Zhao-Qing Feng, Gen-Ming Jin, Jun-Qing Li, and Werner Scheid, Phys. Rev. C 76, 044606 (2007). 2. V. I. Zagrebaev and W. Greiner, Phys. Rev. C 78, 034610 (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, 024606 (2009). 40
Potential energy surface of dinuclear system a- entrance channel; b-fusion channel; c and d are quasifission channels G. Giardina, S. Hofmann, A. I. Muminov, and A. K. Nasirov, Eur. Phys. J. A 8, 205– 216 (2000) Udr (A, Z, , ß 1 , ß 2 ) = B 1 + B 2 + V (A, Z, ß 1 ; ß 2 ; R) - BC N - VC N (L )
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 42
Driving potential Udriving ( c ) for reactions 40 Ar+172 Hf, 86 Kr+130 Xe, 124 Sn+92 Zr leading to formation of compound nucleus 216 Th : Due to peculiarities of shell structure Bfus (Kr) > Bfus (Kr) and, consequently, Udriving=B 1+B 2 -B(1+2)+V( R ) fus (Kr+Xe) < fus(Zr+Sn) 43
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, 024606 (2009). . is capture probability. , which calculated in different theoretical models by different way.
Calculation of the competition between complete fusion and quasifission: Pcn(EDNS, L) Fazio G. et al, Modern Phys. Lett. A 20 (2005) p. 391 46
Map of superheavy elements region 165 173 From paper Yuri Oganessian, Pure Appl. Chem. , Vol. 78, No. 5, pp. 889– 904, 2006 47
Importance of the shell effects in compound nuclei in formation of evaporation residues-superheavy elements (SHE) - Cold fusion A ZX=Cr, Fe, Ni, Zn A ZX +208 Pb, 209 Bi Hot fusion reactions 48 Ca+U, Pu, Cm, Cf 48
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 G. Giardina, et al. Eur. Phys. J. A 8, 205– 216 (2000) * GSI S. Hofmann, Rep. Progr. Phys. 61, 639 (1998);
50
Angular momentum distribution for the complete fusion σfus(L) (Elab) as a function of momentum 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). 51
Driving potential Udriving for reactions 40 Ar+172 Hf, 86 Kr+130 Xe, 124 Sn+92 Zr leading to formation of compound nucleus 216 Th : Udriving=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). 52
Dependence of the intrinsic fusion barrier on angular momentum Udr = B 1 + B 2 - (BCN + VCN (L )) + V (A, Z, ß 1 , α 1 ; ß 2 , α 2 ; R, L) G. Fazio, et al Eur. Phys. J. A 19, 89・ 04 (2004) 53
Fusion hindrance increases by increasing the orbital angular momentum. • F 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. 54
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. 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. C 36, 115 (1987)
Mixing of distribution of fragment masses versus center-of-mass angle W. Q. Shen et al (GSI) Phys. Rev. C 36, 115 (1987)
Main assumptions of the dinuclear system model to study dissipative and quasifission processes.
There are two interacting nuclei – projectile and target which have Z P and ZT protons, and NP and NT neutrons, respectively. The protons and neutrons are placed on the corresponding single-particle states created by the mean-fields UP and UT. The quantum numbers and energies of these single-particle states are found by solving the Schrödinger equation with the Hamiltonians for both nuclei R �� i ZP, MP ZT, MT
Calculation of the physical quantities characterizing multinucleon transfer reactions
Partition of excitation energy between reaction products in heavy ion collisions”, G. G. Adamian, R. V. Jolos, A. K. Nasirov, “ ZEITSCHRIFT FOR PHYSIK A 347, 203 -210 (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 Me. V)+124 Sn, 238 U(1398 Me. V)+124 Sn 56 Fe(505 Me. V)+ 165 Ho, 74 Ge (629 Me. V)+ 165 Ho and 68 Ni(880 Me. V)+ 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… In case of thermal equilibrium we have TP = TT • 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. P T TP > TT
Thermal equilibrium is not reached in fusionfission reactions too ? ! 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. 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. C 36, 115 (1987) 77
Mixing of distribution of fragment masses versus center-of-mass angle W. Q. Shen et al (GSI) Phys. Rev. C 36, 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 Vmin Bqf. L=0 It is important relations between ΔE and difference Ec. m. –Vmin as well as between Ec. m. –Vmin and depth of the potential well Bqf.
Cross sections are found by collision dynamics of projectile and target-nucleus cap (Elab, L; 1 , 2)= (2 L +1) T(Elab, L; 1 , 2) Ldyn and Lmin 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 P’ T Heavy ion collisions T’ Cap ture Deep – inelastic collisions ss Quasifi ( I ) ion ( I I ) P” T” Dinuclear system F 1 F 2 ( I I I ) sion Fast fis Mononucleus Ec. m. =Elab F 1 Co fus mpl ion ete F 2 ( I V) ( V ) g in ol Co Compound nucleus sion s i f n o i Fus ER (ELab , L) = cap(ELab , L) PCN(ELab , L) Wsurv (ELab , L { } Evaporation residues
Capture and fusion cross sections for the 48 Ca+238 U reaction. W. Q. Shen et al Phys. Rev. C 36. 115 (1987) GSI experiment Yu. Ts Oganessian et al, Phys. Rev. C 70 064609 (2004) G. G. Adamian, G. Giardina, A. K. Nasirov, in Cont. of "XIV Int. Workshop on Nuclear Fission" Physics, Obninsk, 1998, Russia, 2000
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 G. Giardina, et al. Eur. Phys. J. A 8, 205– 216 (2000) * GSI S. Hofmann, Rep. Progr. Phys. 61, 639 (1998);
87
Fission products 88
Energy-mass distribution of the reaction products in the 238 U (7. 5 Me. V/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 Aexp of the fragment angular distribution and mean square values of angular momentum from these events. H. Q. Zhang et al, Phys. Rev. C 81, 034611 (2010)
Interpretation of the experimental data presented as fusionfission 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 ? !
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, December, 2008, 124201 96
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 •
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. A 7, 2000, p. 115 -224 99
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, 203 -210 (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 Me. V)+124 Sn, 238 U(1398 Me. V)+124 Sn 56 Fe(505 Me. V)+ 165 Ho, 74 Ge (629 Me. V)+ 165 Ho and 68 Ni(880 Me. V)+ 197 Au is calculated. The results of calculations are in good agreement with the experimental data.
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