Probing Nuclear Structure of Medium and Heavy Unstable Nuclei and Processes with Helium Isotopes

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1 Probing Nuclear Structure of Medium and Heavy Unstable Nuclei and Processes with Helium Isotopes M.K. Gaidarov Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia 1784, Bulgaria University of Aizu-JUSTIPEN-EFES Symposium on "Cutting-Edge Physics of Unstable Nuclei" November 10-13, 2010

2 Terra Incognita

3 Study of densities and charge form factors of light, medium, and heavy neutron-rich exotic nuclei The form factors:

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10 Conclusions (I) The charge form factors are calculated not only in the PWIA but also by solving the Dirac equation for electron scattering in the Coulomb potential of the charge distribution in a given nucleus. The charge distribution in the neutron itself is also taken into account. There is a decrease of the proton densities in the nuclear interior and an increase of its tail at large r with increasing neutron number. The common feature of the charge form factors is the shift of the form factor curves and their minima to smaller values of q with the increase of the neutron number in a given isotopic chain. The theoretical predictions for the charge form factors of exotic nuclei are a challenge for their measurements in the future experiments in GSI and RIKEN and thus, for obtaining detailed information on the charge distributions of such nuclei.

11 Neutron skins in exotic nuclei from Skyrme Hartree-Fock calculations Motivation : New phenomena in nuclei far away from the stability valley Neutron skin formation in neutron-rich nuclei Density dependence of the symmetry energy EOS of asymmetric nuclear matter Theoretical approach : Deformed HF+BCS formalism with Skyrme forces (SLy4) Results : Ni (A=48-78), Kr (A=70-100) and Sn (A= ) isotopes Proton, neutron and charge rms radii Neutron skin thickness from various definitions Neutron skin thickness in deformed nuclei

12 Charge rms radii of tin isotopes r = r + r + ( N / Z ) r + r + r ch p ch p ch n CM SO

13 2 1/2 2 1/2 np n p Δ r = r r Hadron scattering Antiprotonic atoms GDR SDR

14 Helm model ρ 3X ( rr ; ) = Θ ( R r) hs d 3 d 4π Rd Rd = σ = / q 2 3 j q R ln d 1 ( ) 2 1 max 2 qmax Rd qmax F qmax ( ) r 3/2 e 2 2 ( r /2σ ) ρg σ πσ 2 (, ) = ( 2 ) q max q 1 r ur r ur ur ρhe lm r r dr' ( r) = ρ ( ') ( ') hs r ρg F.T. F Hlm e ( q) = F ( q, R ) F ( q, σ ) hs 3 = qr d j 1 d ( qr ) d G e σ 2 q 2 /2

15 N. Fukunishi et al., PRC 48 (1993) 1648 R 1,a R 1,b R 2 R R R 2 1, a 1, b ( R2 ) ( 0) ρ n : = 0.01 ρ n 2 1 ( R ) 1, a ( R ) ρ n : = 4 ρ p 1, a ( R ) ρ p 1, b : = ρ p Δ R = R R ( 0)

16 2 2 Δ Rd = Rd[ n p] Δ R = R + 5σ [ n p] Helm d 2 2 Δ R = R R 5σ [ n p] d[ ] d R n p Δ Helm = d + Δ R = R2 R1, a 2 1/2 Rhs 5/3 r n p [ ] Δ = Δ R = R R 2 1,b

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19 While the profiles of the proton and neutron densities in deformed nuclei change with direction in both oblate and prolate shapes, at the same time, the neutron skin thicknesses remains almost equal along the different directions perpendicular to the surface.

20 Conclusions (II) Theoretical approach based on deformed Skyrme (SLy4) HF+BCS Three isotopic chains (Ni, Kr, Sn) Good agreement with experiment (charge radii) Calculations of proton and neutron density distributions (and radii) Explore: Different definitions of neutron skin thickness Skin formation in isotopic chains Neutron skin along different directions in deformed nuclei Find: Strong dependence of the neutron skin thickness on the various definitions. Linear increase with N While the radial extensions of the densities in deformed nuclei (prolate and oblate) change with the direction, the neutron skin thickness remains practically constant along the different directions

21 Momentum distributions in medium and heavy exotic nuclei Motivation: NMD: key quantity for consistent analysis of the NN correlation effects Importance to study the NMD in exotic nuclei Theoretical approaches: Deformed DDHF+BCS formalism with Skyrme forces (SLy4) Light-front dynamics (LFD) method Local density approximation (LDA) Results: Ni (A=48-78), Kr (A=70-100) and Sn (A= ) isotopes Proton, neutron and total nucleon momentum distributions (NMD s) Total NMD of nuclear matter (NM)

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30 Conclusions (III) Theoretical study based on deformed Skyrme (SLy4) DDHF+BCS method (mean-field), LFD and LDA (correlation methods) The study of the isotopic sensitivity of various kinds of momentum distributions shows different trends: 1) at k 1.5 fm-1 n n (k) increases with the increase of N, whereas n p (k) exhibits an opposite effect. The LFD method does not show this isotopic sensitivity, in contrast to the DDHF+BCS and LDA methods. 2) at low momenta n n (k) decreases while, on the contrary, n p (k) increases with the increase of N. We find that the total momentum distributions of 78 Ni, 86 Kr and 100 Sn isotones (N=50) reveal the same high-momentum tails in all methods. The momentum distributions studied on the example of 98 Kr isotope demonstrate a very weak dependence on the character of deformation. This is valid for all three theoretical approaches.

31 The pairing correlations are shown to influence the highmomentum behavior of n n (k), n p (k) and n(k) in the case of 84 Kr, but the differences between the results with or without BCS correlations are very small. Their effect on the HF momentum distribution is much stronger in the case of NM producing a tail for momenta k k F. The LFD calculations do not show isotopic sensitivity on the obtained NMD s. In the LDA approach some differences between n(k) for protons and neutrons is observed due to Z(N) dependence of the local Fermi momentum k F. The question for the specific values of the parameters β and γ that determine the strength of the correlations is still open. A possible practical way to make predictions for the momentum distributions of exotic nuclei far from the stability line is proposed that provides a systematic description of n(k) in medium-weight and heavy nuclei.

32 A microscopic optical potential approach to 6,8 He+p and 6 He+ 12 C elastic scattering Structure of exotic nuclei Y analyses of their elastic scattering on protons or light targets at different energies 6 He+p: 25.2, 38.3, 41.6 and 71 MeV/N 8 He+p: 15.7, 25.2, 32, 66 and 73 MeV/N 6 He+ 12 C: 3, 38.3 and 41.6 MeV/N He and Li isotopes: 700 MeV/N Phenomenological and microscopic methods: Coordinate-space g-matrix folding method ReOP is microscopically calculated using the folding approach, while the ImOP and the SO terms have been determined phenomenologically

33 The main aim: to calculate dσ/dω of elastic 6,8 He+p and 6 He+ 12 C scattering at energies less than 100 MeV/N studying the possibility to describe the existing experimental data by calculating microscopically not only the ReOP (in a double folding procedure) but also the ImOP (instead of using phenomenological one) within the high-energy approximation (HEA) and using a minimal number of fitting parameters. What we study: the limits of applicability of the HEA OP for different regions of angles ant incident energies the sensitivity of the cross sections to the nuclear densities of 6 He and 8 He the role of the SO interaction and the non-linearity in the calculations of the OP s the nuclear surface effects the role of the renormalization of the depths of ReOP and ImOP the possibility to involve additional physical criteria for a better description of limited number of experimental data

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36 J V =(4π/A) dr r 2 [N R V F (r)] J W =(4π/A) dr r 2 [N R V F (r)] The parameters N R, N I, N R SO and N R SO, the volume integrals J V and J W (in MeV.fm 3 ) as functions of the energy E (in MeV/N) and the total reaction cross sections σ R (in mb) for the 8 He+p scattering in the case of LSSM density.

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38 Cross sections E=38.3A MeV 10 dσ/dσ R θ c.m. [deg] The problem of the ambiguity of the values of N s arises when the fitting procedure concerns a limited number of experimental data. 6 He+ 12 C elastic scattering 9/16

39 Microscopic OP W(r) = W H (r) or V DF (r) U(r) = N R V DF (r) + in I W(r) + in sf I Wsf (r) In order to take into account the breakup effects caused by the dynamic polarization potential we add surface terms W sf (r) = dw(r) dr = r dw(r) dr = r 2 dw(r) dr dw(r 1) = dr Cross sections are calculated using the DWUCK4 code. P. D. Kunz, E. Rost, Computational Nuclear Physics, Vol.2, p.88 6 He+ 12 C elastic scattering 8/16

40 Cross sections dσ/dσ R 10 1 U(r) = N R V DF (r) + in I W(r) in sf dw(r) I r2 dr E=41.6A MeV dσ/dσ R E=38.3A MeV N R =0.797 N I =0.255 N I sf =0.011 W=W H N R =0.578 N I =0.041 N I sf =0.022 W=V DF θ c.m. [deg] N R =0.932 N I =0.028 N I sf =0.019 N R =0.932 N I =0.204 N I sf = θ c.m. [deg] dσ/dω [mb/sr] sf N R =0.790 N I =0.074 N I =0.002 W=W H sf 10-2 N R =0.725 N I =0.040 N I =0.008 W=V DF θ c.m. [deg] E=3A MeV 6 He+ 12 C elastic scattering 13/16

41 Cross sections Volume integrals: J V = 4π N R V DF (r)r 2 dr A p A t J W = 4π N I W(r)r 2 dr. A p A t U = N R V DF + i N I W Physical constraints: J V decrease with the energy increase; J W increases at low energies and then saturates. J [MeV fm 3 ] U = N R V DF + i(n I W - N I sf r dw/dr) J V W=W H J V W=V DF J W W=W H J W W=V DF 300 U = N R V DF + i(n I W - N I sf r 2 dw/dr) E [MeV/nucleon] 6 He+ 12 C elastic scattering 10/16

42 Conclusions (IV) 1. The optical potentials and cross sections of 6 He+p (E=25.2, 41.6 and 71 MeV/N), 8 He+p (E=15.7, 26.25, 32, 66 and 73 MeV/N) and 6 He+ 12 C (E=3, 38.3 and 41.6 MeV/N) elastic scattering were calculated and comparison with the available experimental data was performed. The ReOP (V F ) was calculated microscopically using the folding procedure and M3Y effective interaction based on the Paris NN potential. The ImOP (W H ) was calculated within the HEA. Different model densities of protons and neutrons in 6 He and 8 He were used in the calculations: Tanihata, COSMA, LSSM and JCM. Three different combinations of V F, V H and and W H were used for the OP in calculations of the elastic 6 He+p cross sections. The SO contribution to the OP was included in the calculations. The cross sections were calculated by numerical integration of the Schrödinger equation by means of the DWUCK4 code using all interactions obtained (Coulomb plus nuclear optical potential).

43 2. The results show that the LSSM densities of 6 He and 8 He which have more diffuse tails at larger r than the densities based on Gaussians lead to a better agreement with the data for the elastic scattering at different energies. 3. It was shown that, generally, at energies E>25 MeV/N a good agreement with the experimental data for the differential cross sections can be achieved using OP with calculated both V F and W H varying mainly the volume part of the OP neglecting SO contribution. 4. The explanation of the 6,8 He+p cross sections at lower energies (E<25 MeV/N) needs accounting for the effects of the nuclear surface. In this case the use of ImOP of the HEA type is limited. A more successful explanation of the cross section at low energies could be given by inclusion of polarization contributions due to virtual excitations of inelastic and decay channels of the reactions. 5. The study of the density and energy dependence of the effective M3Y NN forces shows small differences between OP s calculated with and without inclusion of the in-medium effect. The difference between the corresponding cross sections appears at larger angles and increases with the energy increase.

44 6. It was shown that the effects of the Jastrow central short-range NN correlations on the OP s and on the shape of differential cross sections are weak. 7. The problem of the ambiguity of the values of the parameters N R, N I, N R SO, and N I SO when the fitting procedure is applied to a limited number of experimental data is considered. A physical criteria imposed in our work on the choice of the values of the parameters N were the known behavior of the volume integrals J V and J W as functions of the incident energy in the interval 0<E inc <100 MeV/N, as well as the values of the total cross section of scattering and reaction. 8. The behavior of the OP in the nuclear periphery is considered in more details. This gives a possibility to make some conclusions about the contributions of the dynamical polarization terms of the OP or, in other words, about the coupled-channel effects. This approach can be used along with other more sophisticated methods like that from the microscopic g-matrix description of the complex optical potential ant others.

45 Papers 1. A.N. Antonov, D.N. Kadrev, M.K. Gaidarov, E. Moya de Guerra, P. Sarriguren, J.M. Udias, V.K. Lukyanov, E.V. Zemlyanaya, G.Z. Krumova, Charge and Matter Distributions and Form Factors of Light, Medium, and Heavy Neutron-Rich Nuclei, Phys. Rev. C72, (2005) 2. P. Sarriguren, M.K. Gaidarov, E. Moya de Guerra, A.N. Antonov, Nuclear Skin Emergence in Skyrme Deformed Hartree-Fock Calculations, Phys. Rev. C76, (2007) 3. M.K. Gaidarov, G.Z. Krumova, P. Sarriguren, A.N. Antonov, M.V. Ivanov, E. Moya de Guerra, Momentum Distributions in Medium and Heavy Exotic Nuclei, Phys. Rev. C80, (2009) 4. K.V. Lukyanov, V.K. Lukyanov, E.V. Zemlyanaya, A.N. Antonov, M.K. Gaidarov, Calculations of 6 He+p Elastic Scattering Cross Sections Using Folding Approach and High-Energy Approximation for the Optical Potential, Eur. Phys. J. A 33, 389 (2007) 5. V.K. Lukyanov, E.V. Zemlyanaya, K.V. Lukyanov, D.N. Kadrev, A.N. Antonov, M.K. Gaidarov, S.E. Massen, Calculations of 8 He+p Elastic Scattering Cross Sections Using a Microscopic Optical Potential, Phys. Rev. C80, (2009) 6. V.K. Lukyanov, D.N. Kadrev, E.V. Zemlyanaya, A.N. Antonov, K.V. Lukyanov, M.K. Gaidarov, 6 He+ 12 C Elastic Scattering Using a Microscopic Optical Potential, Phys. Rev. C82, (2010)

46 Collaborations: A.N. Antonov,, D.N. Kadrev,, M.V. Ivanov Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia 1784, Bulgaria V.K. Lukyanov,, E.V. Zemlyanaya,, K.V. Lukyanov Joint Institute for Nuclear Research, Dubna , Russia G.Z. Krumova University of Ruse, Ruse 7017, Bulgaria E. Moya de Guerra, J.M. Udias Departamento de Fisica Atomica, Molecular y Nuclear, Facultad de Ciencias Fisicas, Universidad Complutense de Madrid, E Madrid, Spain P. Sarriguren Instituto de Estructura de la Materia, CSIC, Serrano 123, E Madrid, Spain S.E. Massen Department of Theoretical Physics, Aristotle University of Thessaloniki, Thessaloniki, Greece

47 Thank you!