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**Coulomb excitation with radioactive ion beams**

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

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**Coulomb excitation theory - the general approach**

r(t) target r (w) = a (e sinh w + 1) t (w) = a/v (e cosh w + w) a = Zp Zt e2 E-1 projectile Solving the time-dependent Schrödinger equation: iħ d(t)/dt = [HP + HT + V (r(t))] (t) with HP/T being 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) n with n as the eigenstates of HP/T leads to a set of coupled equations for the time-dependent excitation amplitudes an(t) iħ dan(t)/dt = mn|V(t)| m exp[i/ħ (En-Em) t] am(t) The transition amplitude bnm are calculated by the (action) integral bnm= iħ-1 ann|V(t)| amm exp[i/ħ (En-Em) t] dt Finally leading to the excitation probability P(InIm) = (2In+1)-1bnm2 Euroschool Leuven – September 2009

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**Coulomb excitation theory - the general approach**

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 + lm VP(El,m) electric multipole-monopole target excitation, + lm VT(El,m) electric multipole-monopole project. excitation, + lm VP(Ml,m) magnetic multipole project./target excitation lm VT(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/Pe 4p/(2l+1) r–(l+1)Ylm(,) · MP/T(El,m) VP/T(Ml,m) = (-1)m ZT/Pe 4p/(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 electric multipole moments of all orders, while angular correlations give also access to magnetic moments Euroschool Leuven – September 2009

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**Nuclear shapes and electric multipole moments**

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: Euroschool Leuven – September 2009

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**Transition rates in the Coulomb excitation process**

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 Strength parameter Chi^lambda measures the l-pole strength of the coupling of states i and f in a collision of theta=pi and xi=0 Orbital integrals Adiabacity parameter Euroschool Leuven – September 2009

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**Strength parameter E2 as function of (Zp,ZT)**

132Sn 70Se 46Ar ZT ||2/B(E2) · e2b2 78Ni Euroschool Leuven – September 2009

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**Orbital integrals R(E2) as function of and **

Ar+Sn 0.4 MeV 0.8 MeV 1.6 MeV R2(E2; , ) =0.0 =0.2 =0.4 =0.8 Euroschool Leuven – September 2009

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**Cross section for Coulomb excitation**

Differential and total cross sections P(sl) Rutherford Euroschool Leuven – September 2009

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**Angular distribution functions for different multipolarities**

dfsl() Plots of differential and total cross sections page 94 and 97 (for electric) and 113,114 (for magnetic) excitation (from Alder & Winther, 1975) Euroschool Leuven – September 2009

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**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: A2/ E2: A4/ E3: A E4: A8/ M1: 1.79 M2: A2/3 2.08 fEl() fMl() Euroschool Leuven – September 2009

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**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 If Ii b(1) b(2) 2nd order: In If Ii b(1) 1st order: Euroschool Leuven – September 2009

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**Second order perturbation theory (cont.)**

If Ii b(1) b(2) 2nd order: In If Ii b(1) 1st order: P(22) often negligible unless direct excitation through if small/forbidden Euroschool Leuven – September 2009

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**Shape coexistence and excited 0+ states**

0+ states can only be excited via an intermediate 2+ state (if(E0) = 0) If Ii In 0+ 2+ E0 E2 02 20 74Kr 0+ 2+ 4+ 6+ 8+ oblate prolate Shape isomer, E0 transition Configuration mixing: | = | o + | p Euroschool Leuven – September 2009

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**Examples of double-step E2 excitations**

0+ states can only be excited via an intermediate 2+ state (if(E0) = 0) If Ii In 0+ 2+ E0 E2 02 20 Euroschool Leuven – September 2009

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**Examples of double-step E2 excitations**

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

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**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

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**Application to double-step (E2) excitations**

Double-step excitations are important if if << in nf P(22) > P(12) 0+ states can only be excited via an intermediate 2+ state (if = 0) P(2) = |02|2 |20|2 p0(q,s,) with p0(q,s,) = 25/4 (|R20|2+|G20|2) with = 1+ 2 and s= 1/(1+ 2) P(2) (q=p, 1=20) 5/4 |02|2 |20|2 4+ states are usually excited through a double-step E2 since the direct E4 excitation is small P(2) = |02|2 |24|2 p4(q,s,) with p4(q,s,) = 25/4 (|R24|2+|G24|2) P(2) (q=p, 1=2 0) 5/14 |02|2 |24|2 In Ii If 0+ 4+ 2+ E2 E4 If Ii In 0+ 2+ E0 E2 Euroschool Leuven – September 2009

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**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 If Ii a(1) a(2) 2nd order: Ij If Ii a(2) Mf reorientation effect: Euroschool Leuven – September 2009

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**Strength of the reorientation effect**

(2+) [b] CM 76Kr on 208Pb sensitive to diagonal matrix elements intrinsic properties of final state: quadrupole moment including sign Euroschool Leuven – September 2009

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**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

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**Quadrupole deformation and sum rules**

Model-independent method to determine charge distribution parameters (Q,d) from a (full) set of E2 matrix elements 01 + 23 22 21 ~ Q2 ~ Q3 cos3d 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

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**Multi-step Coulomb excitation**

q = 45/16 02 Possible if >> 1 (no perturbative treatment) Example : Rotational band in a strongly deformed nucleus: I I+2 I+6 I+4 I+8 I+10 I+14 I+12 E ħ [keV] J (2) [ħ2MeV-1] superdeformed 152Dy Euroschool Leuven – September 2009

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**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)2 B(sl, 0→l) differential integral 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

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**Euroschool Leuven – September 2009**

Summary 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 sections ds(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 increasing adiabacity parameter and multipolarity l is generally smaller for magnetic than 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 Euroschool Leuven – September 2009

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LLNL-PRES-654342 This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344.

LLNL-PRES-654342 This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344.

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