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Nuclear processes in the continuum

Nuclear processes in the continuum. THE CONTINUUM SHELL MODEL (CSM). The standard formalism to study processes in the nuclear continuum is the Continuum Shell Model. The basis of this formalism is provided by the unity operator written on the real energy axis, i. e.

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Nuclear processes in the continuum

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  1. Nuclear processes in the continuum

  2. THE CONTINUUM SHELL MODEL (CSM) The standard formalism to study processes in the nuclear continuum is the Continuum Shell Model. The basis of this formalism is provided by the unity operator written on the real energy axis, i. e. Where ωn(r) is the wave function of the nth bound state and u(r,E) is the scattering wave at energy E. One discretizes the integral to obtain the CSM representation.

  3. The CSM was used, for instance, to study the halo structure in 11Li (Bertsch et. al., PRC 56, 3054 (1997)). By applying the Cauchy theorem Berggren extended the unity to the complex energy plane,

  4. Notice that neither the sum nor the integral include complex conjugates  Complex probabilities.

  5. The resonances are stationary wave functions with outgoing boundary conditions, where

  6. Neglecting the contribution from the continuum one gets the unity as Where n are bound states and resonances. This gave rise to the first application of the Berggren representation, the Resonant RPA (Curutchet, Vertse, RJL, PRC 39, 1020 (1989)).

  7. Meaning of the complex quantities The energy is E=Er-iΓ/2. The wave function becomes And the probability is Which gives a mean life But this has meaning only if the resonance is narrow, in which case all the imaginary parts are small and, e. g., |φ(r)|2 ≈ Re(φ(r)2)

  8. Evaluation of heavy Li isotopes. Three-neutron nucleus Core 9Li6. Single particle states: 10Li7 (Ψ(r)eikr). E [MeV] k [1/fm] 1s1/2 -0.025 (0,-0.033) 0p1/2 (0.240,-0.064) (0.103,-0.013) 0d5/2 (2.731,-0.545) (0.201,-0.091) 0d3/2 (6.396,-4.898) (0.628,-0.342)

  9. The best way to see whether a resonance has physical meaning is by looking at the wave function. For instance The neutron state 0d3/2 in 10Li7 at (6.396,-4,898) MeV

  10. Instead, the state 1g7/2 in 169Sn119 at (1.667,2.8x10-13) MeV i.e. T=2.4x10-9 sec

  11. Going back to the Berggren representation, to include the scattering waves (i. e.the proper continuum) one has to Discretize the integral Defining the set of vectors {|φ>} such that for the bound and resonant states it is φn(r) = <r|φn> and for the scattering States φp(r) = (hp)1/2 h(r,Ep) one gets It was shown that the set {|n>} forms a representation, called the Berggren representation (RL, Maglione, Sandulescu, Vertse, PLB 367, 1 (1996)). This means that

  12. all processes in the complex energy plane can be described by using the Berggren representation. In particular a shell model in the complex energy plane (CXSM) to describe many-body resonances was formulated (IdBetan, RL, Sandulescu Vertse, PRL 89, 042501 (2002)). The CXSM was applied to study the halo structure of 11Li8 by assuming that the 3 protons form the inert core 9Li6 and that the halo is induced by the two neutrons outside this core (Esbensen, Bertsch, Hencken, PRC56, 3056 (1997)). The single particle states are the resonance 0p1/2 at (0.240,-0.064) and the antibound state 1s1/2 at (-0.025,0) (Zhukov et al, Phys. Rep. 231, 151 (1993)).

  13. To include the antibound state one has to use the integration contour E(ant)=-0.050 MeV, E(p1/2)=(0.24,-0.06) MeV (IdBetan, RL, Sandulescu, Vertse, Wyss, PRC72, 054322 (2005)), The antibound state is very close to the continuum threshold. The wave function looks like follows:

  14. Full line antibound, dashed line bound at the same energy. E=-0.025 MeV

  15. For the 0p1/2 resonance at (0.240,-0.064) MeV the wave function is

  16. We evaluated 11Li by using the CXSM as a two-neutron system. There are two probable physically meaningful states 0+. The corresponding wave functions are where

  17. The 11Li8(gs) (two-particle) radial wave function at r1=r2=r and S=0 (singlet) is E=-0.295 (bound) ≈50% s-state and 50% p-state

  18. For the second 0+ the wave function is

  19. Very recently the ground state and an excited state in 12Li9 (three neutrons outside the core) were measured. The ground state is an antibound state and the excited state is a wide resonance lying at 1.5 MeV and is about 1 MeV wide. The spin of this three-particle resonance is probably 5/2+ (Roger et. al., to be published). To analyse this nucleus we used the Multistep Shell Model in the complex energy plane (CXMSM). In this model the basis set of states have consists of the tensorial product of previously evaluated states. In our case the core is and the basis states are of the form Or, schematically, The three-body basis is very large because there are many scattering waves. However only very few of these states are relevant.

  20. Within the CXMSM basis one can decide the relevant one- and two-particle states by looking at the corresponding wave functions. For the single-particles we have seen that the 1s1/2 antibound state and that the 0p1/2 resonance are fundamental to describe 11Li. But Also the 0d5/2 resonance may be physically relevant.

  21. For the two-particle states we saw that 11Li(gs) is bound. The only other state to be relevant was found to be the state 21+.

  22. With this small basis set of states we evaluated 12Li Zhen Xiang Xu, RL, Chong Qi, T. Roger, P. Roussel-Chomaz, H. Savajols and R. Wyss. To be published

  23. Summary The treatment of processes occurring in the continuum part of the nuclear spectrum may require time dependent formalisms. Alternatively, one may extend usual (e. g. shell model) formalisms to the energy plane. For this one has to pay the price of dealing with complex energies and probabilities. To decide whether a evaluated resonant state is physically meaningful one can examine the corresponding wave function. A resonance appears because there is a barrier that traps the system inside the nucleus. Therefore the wave function inside the nucleus has to be real and localized. This formalism can be applied to study many body resonances just applying the same methods developed for bound states but expended to the complex energy plane.

  24. neutrons protons F. Janouch and RJL, PRC 25, 2123 (1982)

  25. F. Janouch and RJL, PRC 27, 896 (1983)

  26. ___________________________________ ______________________________ ______ ______ ______ ______ G. Dodig-Crnkovic, F. A. Janouch and RJL, PLB139, 143 (1984)

  27. G. Dodig-Crnkovic, F. A. Janouch, RJL and L. J. Sibanda, NPA 444, 419 (1985) K. Varga, R. G. Lovas and RJL, PRL 69, 37 (1992) R. G. Lovas, RJL, A. Insolia, K. Varga and D. S. Delion, Phys. Rep. 294, 265 (1998)

  28. ______________________________________________________ _________ _________ ______________________________ __________ M. W. Herzog, RJL and L. J. Sibanda, PRC, 259 (1985)

  29. P. Curutchet, T. Vertse and RJL, PRC39, 1020 (1989) T. Vertse, E. Maglione and RJL, NPA584, 13 (1995)

  30. V(r) = 0, r > R J. Bang, F. A. Gareev, M. H. Gizzatkulov and S. A. Goncharov, Nucl. Phys. A309, 381 (1978)

  31. RJL, E. Maglione, N. Sandulescu and T. Vertse, PLB367, 1 (1996)

  32. R. Id Betan, RJL, N. Sandulescu and T. Vertse, PRL89, 042501 (2002)

  33. Resonant HF Binding energies A. T. Kruppa, P.-H. Heenen, H. Flocard and RJL, PRL, 79, 2217 (1997)

  34. Binding energy (E) and root mean square radius (r2) in different self-consistent mean-field models for the N=20 isotones. 42Ti 44Cr 46Fe -E r2-E r2-E r2 HFB 350.84 3.50 354.39 3.58 355.51 3.65 HF-BCS-B 350.52 3.50 353.95 3.57 355.15 3.64 HF-BCS-R1 350.65 3.51 354.13 3.58 355.28 3.64 HF-BCS-R2 350.90 3.51 354.44 3.58 355.51 3.65 A. T. Kruppa, P. H. Heenen and RJL, PRC 63, 044324 (2001)

  35. H. J. Unger, NPA 164, 564 (1967) N. Sandulescu, N. Van Giai and RJL, PRC 61(R), 061301 (2000)

  36. bound antibound R. Id Betan, RJL, N. Sandulescu, T. Vertse and R. Wyss, PRC 72, 054322 (2005)

  37. M. Grasso, N. Sandulescu, N. Van Giai and RJL, PRC 64, 064321 (2001)

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