Coulomb excitation with radioactive ion beams

1
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
2
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

• Solving the time-dependent Schrödinger
  equation
• ih 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)
• ih dan(t)/dt ?m??nV(t) ?m? expi/h (En-Em)
  t am(t)
• The transition amplitude bnm are calculated by
  the (action) integral
• bnm ih-1 ? ?an?nV(t) am?m? expi/h (En-Em)
  t dt
• Finally leading to the excitation probability
• P(In?Im) (2In1)-1bnm2

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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,slgt0) higher order
  multipole-multipole terms (small)
• VP/T(El,m) (-1)m ZT/Pe 4p/(2l1)
  r(l1)Ylm(?,?) MP/T(El,m)
• VP/T(Ml,m) (-1)m ZT/Pe 4p/(2l1) i/cl
  r(l1)dr/dtLYl,m(?,?) MP/T(Ml,m)
• electric multipole moment
• M(El,m) ? r(r) rl Ylm(r) d3r
• magnetic multipole moment
• M(Ml,m) -i/c(l1) ? j(r) rl (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

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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

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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
Orbital integrals
Adiabacity parameter
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Strength parameter ?E2 as function of (Zp,ZT)
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Orbital integrals R(E2) as function of ? and ?
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Cross section for Coulomb excitation
Differential and total cross sections
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Angular distribution functions for different
multipolarities
dfsl(?)
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Total cross sections for different multipolarities
B(sl) values for single particle like
transitions (W.u.) Bsp(l) (2l1)
9e2/4p(3l)-2 R2l x 10(hc/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(?)
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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

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Second order perturbation theory (cont.)
P(22) often negligible unless direct excitation
through ?i?f small/forbidden
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Shape coexistence and excited 0 states

• 0 states can only be excited via an intermediate
  2 state (?if(E0) 0)

oblate
prolate
Shape isomer, E0 transition Configuration mixing
? ? ? o ? ? p ?
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Examples of double-step E2 excitations

• 0 states can only be excited via an intermediate
  2 state (?if(E0) 0)

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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

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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
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Application to double-step (E2) excitations

• Double-step excitations are important if ?if ltlt
  ?in ?nf ? P(22) gt P(12)
• 0 states can only be excited via an intermediate
  2 state (?if 0)
• ? P(2) ?0?22 ?2?02 p0(q,s,?) with
  p0(q,s,?) 25/4 (R202G202)
• with ? ?1 ?2 and s ?1/(?1 ?2)
• P(2) (qp, ?1?2?0) ? 5/4 ?0?22 ?2?02
• 4 states are usually excited through a
  double-step E2 since the direct E4 excitation is
  small
• ? P(2) ?0?22 ?2?42 p4(q,s,?) with
  p4(q,s,?) 25/4 (R242G242)
• P(2) (qp, ?1?2 ? 0) ? 5/14 ?0?22 ?2?42

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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

reorientation effect
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Strength of the reorientation effect
sensitive to diagonal matrix elements ? intrinsic
properties of final state quadrupole moment
including sign
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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 Igt½ states
M. Girod, CEA
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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
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

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Multi-step Coulomb excitation
Possible if ? gtgt 1 (no perturbative
treatment) Example Rotational band in a
strongly deformed nucleus
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Coulomb excitation the different energy regimes
Low-energy regime (lt 5 MeV/u)
High-energy regime (gtgt5 MeV/u)
Energy cut-off
Spin cut-off Lmax up to 30? mainly single-step
excitations
Cross section d?/d? ?IiM(sl)If? ?l
(Zpe2/ hc)2 B(sl, 0?l) differential integral
Luminosity low mg/cm2 targets high g/cm2
targets Beam intensity high gt103 pps low a
few pps
Comprehensive study of low-lying exitations
First exploration of excited states in very
exotic nuclei
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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