Coulomb excitation with radioactive ion beams

1. Motivation and introduction Theoretical aspects of Coulomb excitation Experimental considerations, set-ups and analysis techniques Recent highlights and future perspectives Coulomb excitation with radioactive ion beams Lecture given at the Euroschool 2009 in Leuven Wolfram KORTEN CEA Saclay Euroschool Leuven – September 2009

2. Solving the time-dependent Schrödinger equation: iħ d(t)/dt = [HP + HT +V(r(t))] (t) with HP/Tbeing the free Hamiltonian of the projectile/target nucleus and V(t) being the time-dependent electromagnetic interaction (remark: often only target or projectil excitation are treated) Expanding (t) = n an(t) nwith nas the eigenstates of HP/T leads to a set of coupled equations for the time-dependent excitation amplitudes an(t) iħ dan(t)/dt = mn|V(t)| m exp[i/ħ (En-Em) t] am(t) The transition amplitude bnm are calculated by the (action) integral bnm= iħ-1  ann|V(t)| amm exp[i/ħ (En-Em) t] dt Finally leading to the excitation probability P(InIm) = (2In+1)-1bnm2 Coulomb excitation theory - the general approach b target r (w) = a (e sinh w + 1) t (w) = a/v (e cosh w + w) a = Zp Zt e2 E-1 r(t) projectile Euroschool Leuven – September 2009

3. The coupled equations for an(t) are usually solved by a multipole expansion of the electromagnetic interaction V(r(t)) VP-T(r) = ZTZPe2/r monopole-monopole (Rutherford) term + lmVP(El,m)electric multipole-monopole target excitation, + lmVT(El,m) electric multipole-monopole project. excitation, + lmVP(Ml,m) magnetic multipole project./target excitation + lmVT(Ml,m) (but small at low v/c) + O(sl,s’l’>0)higher order multipole-multipole terms (small) VP/T(El,m)= (-1)m ZT/Pe4p/(2l+1) r–(l+1)Ylm(,)· MP/T(El,m) VP/T(Ml,m) = (-1)m ZT/Pe4p/(2l+1) i/cl r–(l+1)dr/dtLYl,m(,)· MP/T(Ml,m) electric multipole moment: M(El,m) =  r(r‘) r‘l Ylm(r‘)d3r‘ magnetic multipole moment: M(Ml,m) = -i/c(l+1)  j(r‘) r‘l (ir)Yl,m(r‘)d3r‘ Coulomb excitation cross section is sensitive to electricmultipole moments of all orders, while angular correlations give also access to magnetic moments Coulomb excitation theory - the general approach Euroschool Leuven – September 2009

4. Electric multipole moments can be linked to Deformation parameters of the nuclear mass distribution For axially symmetric shapes (bl = al0) and a homogenous density distribution r the quadrupole, octupole and hexadecupole moments (Q2,Q3,Q4) become: Nuclear shapes and electric multipole moments Euroschool Leuven – September 2009

5. 1st order perturbation theory applicable if only one state is excited, e.g. 0+2+ excitation, and for small excitation probability (e.g. semi-magic nuclei)  1st order transition probability for multipolarity l Transition rates in the Coulomb excitation process Strength parameter Orbital integrals Adiabacity parameter Euroschool Leuven – September 2009

6. 46Ar 78Ni 70Se ||2/B(E2) · e2b2 132Sn ZT Strength parameter E2 as function of (Zp,ZT) Euroschool Leuven – September 2009

7. R2(E2; , ) =0.0 Ar+Sn =0.2 0.4 MeV =0.4 =0.8 0.8 MeV  1.6 MeV Orbital integrals R(E2) as function of  and  Euroschool Leuven – September 2009

8. Rutherford P(sl) Cross section for Coulomb excitation Differential and total cross sections Euroschool Leuven – September 2009

9. Angular distribution functions for different multipolarities dfsl() Euroschool Leuven – September 2009

10. Total cross sections for different multipolarities B(sl) values for single particle like transitions (W.u.): Bsp(l) = (2l+1) 9e2/4p(3+l)-2 R2l x 10(ħc/MpR0)2 B(sl) [e2bl] 208Pb E1: 6.45 10-4 A2/3 2.3 10-2 E2: 5.94 10-6 A4/3 7.3 10-3 E3: 5.94 10-8 A2 2.6 10-3 E4: 6.28 10-10A8/3 9.5 10-4 M1: 1.79 M2: 0.0594 A2/3 2.08 fEl() fMl() Euroschool Leuven – September 2009

11. If In b(1) If Ii In b(2) b(1) b(2) Ii 2nd order: 1st order: Transition rates in the Coulomb excitation process • Second order perturbation theory becomes necessary if several states can be excited from the ground state or when multiple excitations are possible i.e. for larger excitation probabilities  2nd order transition probability for multipolarity l Euroschool Leuven – September 2009

12. If In b(1) If Ii In b(2) b(1) b(2) Ii 2nd order: 1st order: Second order perturbation theory (cont.) P(22) often negligible unless direct excitation through if small/forbidden Euroschool Leuven – September 2009

13. 0+ states can only be excited via an intermediate 2+ state (if(E0) = 0) 2+ In 20 E2 If 0+ 74Kr 02 E0 E2  Ii 0+ 6+ 8+ 6+ 4+ 4+ 2+ 2+ 0+ 0+  Shape coexistence and excited 0+ states oblate prolate Shape isomer, E0 transition Configuration mixing: |  =  | o +  | p  Euroschool Leuven – September 2009

14. 0+ states can only be excited via an intermediate 2+ state (if(E0) = 0) 2+ In 20 E2 If 0+ 02 E0 E2 Ii 0+ Examples of double-step E2 excitations Euroschool Leuven – September 2009

15. 4+ states can be excited through a double-step E2 or a direct E4 excitation quadrupole hexadecapole + Double E2 Direct E4 4+ If E2 In 2+ E4 E2 Ii 0+ Examples of double-step E2 excitations Euroschool Leuven – September 2009

16. Double-step E2 vs. E4 excitation of 4+ states p4 and d functions for different scattering angles and 1- 2 ratios Euroschool Leuven – September 2009

17. Mf Ij If If a(2) Ij a(2) a(1) a(2) Ii Ii 2nd order: The reorientation effect • Specific case of second order perturbation theory where the „intermediate“ states are the m substates of the state of interest  2nd order excitation probability for 2+ state reorientation effect: Euroschool Leuven – September 2009

18. 76Kr on 208Pb (2+) [b] CM Strength of the reorientation effect sensitive to diagonal matrix elements  intrinsic properties of final state: quadrupole moment including sign Euroschool Leuven – September 2009

19. Quadrupole deformation of nuclear ground states Coulomb excitation can, in principal, map the shape of all atomic nuclei:  Quadrupole (and higher-order multipole moments) of I>½ states M. Girod, CEA Euroschool Leuven – September 2009

20. + + + + 01 23 22 21 Quadrupole deformation and sum rules Model-independent method to determine charge distribution parameters (Q,d) from a (full) set of E2 matrix elements ~ Q3 cos3d ~ Q2 • ground state shape can be determined by a full set of E2 matrix elements i.e. linking the ground state to all collective 2+ states Euroschool Leuven – September 2009

21. I+14 I+12 I+10 I+8 E I+6 J (2) [ħ2MeV-1] I+4 superdeformed 152Dy I+2 q = 45/16 02 I ħ [keV] Multi-step Coulomb excitation Possible if  >> 1 (no perturbative treatment) Example : Rotational band in a strongly deformed nucleus: Euroschool Leuven – September 2009

22. Coulomb excitation – the different energy regimes Low-energy regime (< 5 MeV/u) High-energy regime (>>5 MeV/u) Energy cut-off Spin cut-off: Lmax: up to 30ћ mainly single-step excitations Cross section: d/d ~ Ii|M(sl)|Ifl~ (Zpe2/ ħc)2B(sl, 0→l) differentialintegral Luminosity: low mg/cm2 targets high g/cm2 targets Beam intensity: high >103 pps low a few pps Comprehensive study of low-lying exitations First exploration of excited states in very “exotic” nuclei Euroschool Leuven – September 2009

23. Coulomb excitation probability P(Ip) increases with increasing strength parameter (), i.e. ZP/T, B(sl), 1/D, qcm decreasing adiabacity parameter (), i.e.DE, a/v Differential cross sectionsds(q)/dW show varying maxima depending on multipolarity l and adiabacity parameter   allows to distinguish different multipolarities (E2/M1, E2/E4 etc.) Total cross section stot decreases with increasingadiabacity parameter  andmultipolarityl is generally smaller for magneticthan for electric transitions Second and higher order effects lead to “virtual” excitations influencing the real excitation probabilities allow to excite 0+ states and to measure static moments lead to multi-step excitations Summary Euroschool Leuven – September 2009