Systematics of the α-decay fine structure in even-even nuclei

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1 Systematics of the α-decay fine structure in even-even nuclei A. Dumitrescu 1,4, D. S. Delion 1,2,3 1 Department of Theoretical Physics, NIPNE-HH 2 Academy of Romanian Scientists 3 Bioterra University 4 Department of Physics, University of Bucharest alexandru.dumitrescu@theory.nipne.ro delion@theory.nipne.ro September 02, /27

2 Outline of our paper 1 We perform systematic calculations for even-even nuclei in the Z > 50, N > 82 region concerning ground-band energy levels quadrupole electric transitions within the ground-band α-decay half-lives and intensities of α-transitions to excited states by means of a collective Coherent State Model (CSM) for the description of the daughter nucleus an α-daughter interaction containing a monopole component, estimated using a double folding procedure with M3Y interaction plus a repulsive core, and a quadrupole-quadrople (QQ) component. 1 To be published in Atomic Data and Nuclear Data Tables. 2/27

3 Outline of this presentation 1. α-decay - a probe for nuclear structure 2. Nuclear structure - the coherent state model 3. Coupled channels approach to α-emission 4. Conclusions 3/27

4 α-decay - a probe for nuclear structure nuclear structure A Parent Z 0 + A-4 Daughter Z α2 6 + B screen α-source decay intensities I J +=log 10 (Γ 0 +/Γ J +) I I J + depends linearly on E J + N<126 N> E + 2 (kev) 4/27

5 The structure of even-even α-emitters Most even-even α-emitters have the low-level structure interpreted in terms of collective surface dynamics, ranging from harmonic vibrations to rigid rotations MeV MeV vibrational spectrum rotational spectrum 5/27

6 Unified description of vibrational, transitional and rotational nuclei - coherent states 3, 4 The intrinsic ground state of an axially deformed nucleus is a coherent superposition of quadrupole phonons: ( ) ψ g = e d b 20 b 20 0 The deformation parameter is proportional to the static quadrupole deformation 2, thus indicating the spherical, prolate or oblate shape of the nucleus: d = κβ 2 2 Tabulated in P. Möller and J.R. Nix, Nucl. Phys. A 272, 502 (1995). 3 P. O. Lipas and J. Savolainen, Nuclear Physics A 130, 77 (1969). 4 A. A. Raduta et al., Nuclear Physics A 381, 253 (1982). 6/27

7 Relation between the CSM & quadrupole deformation parameters 6 5 N<126, σ 2 <10% N<126, σ 2 >10% N>126, σ 2 <10% d β 2 7/27

8 Ground-band energy levels States in the laboratory frame are obtained through angular momentum projection: J = N (g) J ˆP M0 J ψ g ϕ (g) The simplest approximation for the energy spectrum uses a harmonic Hamiltonian: E J (d) = A 1 [ ϕ (g) J ˆN ϕ (g) J ϕ (g) 0 ˆN ϕ (g) 0 ] 8/27

9 Energy spectrum E J /A d 9/27

10 Hamiltonian Strength Parameter versus the CSM Deformation N<126, σ 2 <10% N<126, σ 2 >10% N>126, σ 2 <10% A 1 (kev) d 10/27

11 Ratio of energy levels versus the CSM deformation parameter for J = E 4 +/E N<126, σ 2 <10% N<126, σ 2 >10% N>126, σ 2 <10% d 11/27

12 Ratio of energy levels versus the CSM deformation parameter for J = E 12 +/E N<126, σ 2 <10% N>126, σ 2 <10% d 12/27

13 Coupled channels approach to α-emission Ground to excited state α-decay process: P D(J)+α Wavefunction of the α-particle + core system: Ψ(b 2,R) = f J (R) R Z J(b 2,Ω) J [ ] Z J (b 2,Ω) ϕ (g) J (b 2 ) Y J (Ω) b 2 designates the quadrupole degrees of freedom of the core R is the relative distance between the core and α-particle Ω is the orientation of the α-particle in the laboratory frame 0 13/27

14 The α-core system: a pictorial representation 14/27

15 The α-particle + core Hamiltonian Stationary Schrödinger equation: HΨ(b 2,R) = Q α Ψ(b 2,R) System Hamiltonian, containing the kinetic, core and interaction terms: H = 2 2µ 2 R +H D(b 2 )+V(b 2,R) The α-core potential, features a spherical and QQ interaction : V(b 2,R) = V 0 (R)+C QQ(b 2,R) 15/27

16 U 238 +α-particle interaction potential - the spherical part Oscillator strength: c=50 MeV/fm 2 The height of the Coulomb barrier can be adjusted through the quenching factor v a to reproduce the total half-life. V 0 (MeV) Q α =4.984 MeV 0 The depth of the potential is adjusted so that the first resonance corresponds to the experimental Q-Value R (fm) 16/27

17 The quenching factor versus the logarithm of the reduced decay width to the ground state N<126 N>126 SHE v a log 10 γ /27

18 Logarithm of the total α-decay half-life versus mass number 20 exp pred log 10 T α (s) A 18/27

19 The QQ coupling constant - transitions to excited states The effective coupling constant is proportional to the CSM deformation: C = C 0 (1 ) 2 7 a αd 19/27

20 Boundary Conditions I The radial functions f J must be smooth up to their first derivative. At infinity they fulfill the condition of Gamow resonances, thus behaving like outgoing Coulomb-Hankel waves. From the continuity equation one obtains the total decay width as a sum of partial widths Γ = J Γ J = J v J lim R f J(R) 2 = J v J N J 2, with v J = k J µ and N J is the scattering amplitude obtained from the normalization of the solution. 20/27

21 Boundary Conditions II In conclusion, diagonalizing the core + α-particle Hamiltonian for different values of the coupling constant C yields decay widths that can be compared to experimental values and allow the calculation of the decay intensities I J + = log 10 Γ 0 Γ J. 21/27

22 QQ coupling strength versus CSM deformation 0.6 N<126 N> C f J f J R (fm) J=0 J= R (fm) J=0 J= d 22/27

23 I 2 + in the Th-Cf region Th U Pu Cm exp pred Cf 0.8 I n 23/27

24 I 4 + in the Th-Cf region Th U Pu Cm exp pred Cf I n 24/27

25 I 6 + in the Th-Cf region Th U Pu Cm exp pred Cf I n 25/27

26 Conclusions The simplest version of the CSM formalism allows a unified description of the structure of vibrational, transitional and well-deformed nuclei and is suited for the study of even-even α-emitters. The coupled channels method gives a complete description of the α-decay spectrum, the decay intensity to excited states following from a QQ coupling strength depending linearly on the CSM deformation. The disagreement between theory and experiment for the I J + values in the Pu region is still under investigation, but so far only correlations with other microscopic observables have been discovered 5. 5 For the connection between the peak of I + 4 in the Pu region and the two-neutron separation energy near the subshell N=142, see D. Bucurescu, N.V. Zamfir, Phys. Rev. C 87, (2013). 26/27

27 Thank you! 27/27

COUPLED CHANNELS FORMALISM

Annals of the Academy of Romanian Scientists Online Edition Physics Series ISSN 2066 8589 Volume 5, Number 1/2015 5 FINE STRUCTURE OF α-transitions WITHIN THE COUPLED CHANNELS FORMALISM D.S. DELION 1,