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
Euroschool Leuven – September 2009

2. 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 = 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
Euroschool Leuven – September 2009

3. 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

4. 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

5. 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

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

7. 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

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

9. 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

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

11. 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

12. 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 if small/forbidden
Euroschool Leuven – September 2009

13. 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
02
20
74Kr   0+ 2+ 4+ 6+ 8+
oblate
prolate
Shape isomer, E0 transition
Configuration mixing: |  =  | o  +  | p 
Euroschool Leuven – September 2009

14. 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
02
20
Euroschool Leuven – September 2009

15. 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

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. 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) = |02|2 |20|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=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 p4(q,s,) with p4(q,s,) = 25/4 (|R24|2+|G24|2)
P(2) (q=p, 1=2  0)  5/14 |02|2 |24|2 In Ii If 0+ 4+ 2+ E2 E4 If Ii In 0+ 2+ E0 E2
Euroschool Leuven – September 2009

18. 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

19. 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

20. 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

21. 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

22. 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
ħ [keV]
J (2) [ħ2MeV-1]
superdeformed 152Dy
Euroschool Leuven – September 2009

23. 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

24. 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