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Wolfram KORTEN 1 Euroschool Leuven – September 2009 Coulomb excitation with radioactive ion beams Motivation and introductionMotivation and introduction.

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Presentation on theme: "Wolfram KORTEN 1 Euroschool Leuven – September 2009 Coulomb excitation with radioactive ion beams Motivation and introductionMotivation and introduction."— Presentation transcript:

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

2 Wolfram KORTEN 2 Euroschool Leuven – September 2009 Coulomb excitation theory - the general approach Solving the time-dependent Schrödinger equation: iħ d  (t)/dt = [H P + H T + V (r(t))]  (t) with H P/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 a n (t)  n with  n as the eigenstates of H P/T leads to a set of coupled equations for the time-dependent excitation amplitudes a n (t) iħ da n (t)/dt =  m  n |V(t)|  m  exp[i/ħ (E n -E m ) t] a m (t) The transition amplitude b nm are calculated by the (action) integral b nm = i ħ -1   a n  n |V(t)| a m  m  exp[i/ħ (E n -E m ) t] dt Finally leading to the excitation probability P(I n  I m ) = (2I n +1) -1 b nm 2 b projectile r(t) target r (w) = a (  sinh w + 1) t (w) = a/v  (  cosh w + w) a = Z p Z t e 2 E -1

3 Wolfram KORTEN 3 Euroschool Leuven – September 2009 Coulomb excitation theory - the general approach The coupled equations for a n (t) are usually solved by a multipole expansion of the electromagnetic interaction V(r(t)) V P-T (r) = Z T Z P e 2 /rmonopole-monopole (Rutherford) term +    V P (  ) electric multipole-monopole target excitation, +    V T (  ) electric multipole-monopole project. excitation, +    V P (  ) magnetic multipole project./target excitation +    V T (  )(but small at low v/c) + O(  ’ ’>0)higher order multipole-multipole terms (small) V P/T (  ) =   Z T/P e   /(2  r –( +1) Y  ( ,  ) · M P/T (  V P/T (  ) =   Z T/P e   /(2 +1) i/c  r –( +1) dr/dtLY  ( ,  ) · M P/T (  electric multipole moment: M(E   (r‘) r‘ Y  (r‘)  d 3 r‘ magnetic multipole moment: M(M  -i/c(  j(r‘) r‘ l (ir  )Y  (r‘)  d 3 r‘  Coulomb excitation cross section is sensitive to electric multipole moments of all orders, while angular correlations give also access to magnetic moments

4 Wolfram KORTEN 4 Euroschool Leuven – September 2009 Nuclear shapes and electric multipole moments Electric multipole moments can be linked to Deformation parameters of the nuclear mass distribution For axially symmetric shapes (     and a homogenous density distribution  the quadrupole, octupole and hexadecupole moments (Q 2,Q 3,Q 4 ) become:

5 Wolfram KORTEN 5 Euroschool Leuven – September 2009 Transition rates in the Coulomb excitation process 1 st 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)  1 st order transition probability for multipolarity  Orbital integrals Adiabacity parameter Strength parameter

6 Wolfram KORTEN 6 Euroschool Leuven – September 2009 Strength parameter  E2 as function of (Z p,Z T ) 132 Sn 70 Se 46 Ar ZTZT |  | 2 /B(E2) · e 2 b 2 78 Ni

7 Wolfram KORTEN 7 Euroschool Leuven – September 2009 Orbital integrals R(E2) as function of and  R 2 (E2;,  )  =0.0  =0.2  =0.4  =0.8 Ar+Sn 0.4 MeV 0.8 MeV 1.6 MeV

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

9 Wolfram KORTEN 9 Euroschool Leuven – September 2009 Angular distribution functions for different multipolarities df  (  )

10 Wolfram KORTEN 10 Euroschool Leuven – September 2009 Total cross sections for different multipolarities B(  ) values for single particle like transitions (W.u.): B sp ( ) = (2 +1) 9e 2 /4   R 2  x 10(ħc/M p R 0 ) 2 B(  ) [e 2 b ] 208 Pb E1: A 2/ E2: A 4/ E3: A E4: A 8/ M1: 1.79 M2: A 2/ f E (  )f M (  )

11 Wolfram KORTEN 11 Euroschool Leuven – September 2009 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  2 nd order transition probability for multipolarity  Transition rates in the Coulomb excitation process IfIf IiIi b (1) b (2) 2 nd order: InIn InIn IfIf IiIi b (1) 1 st order:

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

13 Wolfram KORTEN 13 Euroschool Leuven – September 2009 Shape coexistence and excited 0 + states oblateprolate Shape isomer, E0 transition Configuration mixing: |   =  | o  +  | p  74 Kr   IfIf IiIi InIn E0 E2 0202 2020 0 + states can only be excited via an intermediate 2 + state (  if (E0) = 0)

14 Wolfram KORTEN 14 Euroschool Leuven – September 2009 Examples of double-step E2 excitations 0 + states can only be excited via an intermediate 2 + state (  if (E0) = 0) IfIf IiIi InIn E0 E2 0202 2020

15 Wolfram KORTEN 15 Euroschool Leuven – September 2009 Examples of double-step E2 excitations 4 + states can be excited through a double-step E2 or a direct E4 excitation InIn IiIi IfIf E2 E4 Double E2Direct E4 quadrupolehexadecapole +

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

17 Wolfram KORTEN 17 Euroschool Leuven – September 2009 Application to double-step (E2) excitations Double-step excitations are important if  if P (12)  0 + states can only be excited via an intermediate 2 + state (  if = 0)  P (2) = |  0  2 | 2 |  2  0 | 2   ,s,  ) with   ,s,  ) = 25/4 (|R 20 | 2 +|G 20 | 2 ) with  =  1 +  2 and s=  1 /(  1 +  2 ) P (2) (  1 =  2  0)  5/4 |  0  2 | 2 |  2  0 | 2  4 + states are usually excited through a double-step E2 since the direct E4 excitation is small  P (2) = |  0  2 | 2 |  2  4 | 2   ,s,  ) with   ,s,  ) = 25/4 (|R 24 | 2 +|G 24 | 2 ) P (2) (  1 =  2  0)  5/14 |  0  2 | 2 |  2  4 | 2 IfIf IiIi InIn E0 E2 InIn IiIi IfIf E4

18 Wolfram KORTEN 18 Euroschool Leuven – September 2009 The reorientation effect Specific case of second order perturbation theory where the „intermediate“ states are the m substates of the state of interest  2 nd order excitation probability for 2 + state  IfIf IiIi a (1) a (2) 2 nd order: IjIj IjIj reorientation effect: IfIf IiIi a (2) MfMf

19 Wolfram KORTEN 19 Euroschool Leuven – September 2009 Strength of the reorientation effect  (2 + ) [b]  CM 76 Kr on 208 Pb sensitive to diagonal matrix elements  intrinsic properties of final state: quadrupole moment including sign

20 Wolfram KORTEN 20 Euroschool Leuven – September 2009 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

21 Wolfram KORTEN 21 Euroschool Leuven – September 2009 Quadrupole deformation and sum rules Model-independent method to determine charge distribution parameters (Q,  from a (full) set of E2 matrix elements  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 ~ Q 2 ~ Q 3 cos3 

22 Wolfram KORTEN 22 Euroschool Leuven – September 2009 Multi-step Coulomb excitation q =  45/16  0  2 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E ħ  [keV] J (2) [ħ 2 MeV -1 ] superdeformed 152 Dy

23 Wolfram KORTEN 23 Euroschool Leuven – September 2009 Coulomb excitation – the different energy regimes Low-energy regime (< 5 MeV/u) High-energy regime (>>5 MeV/u) Energy cut-off Spin cut-off:L max :up to 30ћmainly single-step excitations Cross section: d  /d  ~  I i |M(  )|I f   ~ (Z p e 2 / ħc) 2 B( , 0→ ) differential integral Luminosity: low mg/cm 2 targetshigh g/cm 2 targets Beam intensity:high >10 3 ppslow a few pps Comprehensive study of low-lying exitations First exploration of excited states in very “exotic” nuclei

24 Wolfram KORTEN 24 Euroschool Leuven – September 2009 Summary Coulomb excitation probability P(I  ) increases with increasing strength parameter (  ), i.e. Z P/T,  ), 1/D,  cm decreasing adiabacity parameter (  ),i.e.  E, a/v  Differential cross sections  d  /d  show varying maxima depending on multipolarity and adiabacity parameter   allows to distinguish different multipolarities (E2/M1, E2/E4 etc.) Total cross section  tot decreases with increasing adiabacity parameter  and multipolarity 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


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