1. Hans-Jürgen Wollersheim
Coulomb excitation
Hans-Jürgen Wollersheim γ
target excited
projectile
Beam
Target
HPGe θ

2. Hans-Jürgen Wollersheim
Coulomb excitation
Hans-Jürgen Wollersheim
Nuclear excitation by electromagnetic field acting between nuclei.
observables:
1. scattering angle
2. intensity +
𝜗 𝑙𝑎𝑏 ⟹ 𝜃 𝑐𝑚 b
projectile
target +
𝑎∙𝑐𝑜𝑡 𝜃 𝑐𝑚 2 =
𝑑 𝜎 𝑅𝑢𝑡ℎ 𝑑 Ω 𝑐𝑚 = 𝑎 2 4 ∙ 𝑠𝑖𝑛 −4 𝜃 𝑐𝑚 2
~10-22s
𝑑 𝜎 𝑖𝑛𝑒𝑙 𝑑 Ω 𝑐𝑚 = 𝒂 𝒊⟶𝒇 2 ∙ 𝑑 𝜎 𝑅𝑢𝑡ℎ 𝑑 Ω 𝑐𝑚
α-particle spectroscopy
ϑlab
152Sm target
Si-detector
α-particles
inelastic scattering: kinetic energy is transferred into nuclear excitation energy
H.J. Wollersheim et al., Phys. Lett 48B (1974) 323

3. Coulomb excitation +  + If Ii
Nuclear excitation by electromagnetic field acting between nuclei.
observables:
1. scattering angle
2. intensity +
𝜗 𝑙𝑎𝑏 ⟹ 𝜃 𝑐𝑚 b
projectile
target +
𝑎∙𝑐𝑜𝑡 𝜃 𝑐𝑚 2 =
𝑑 𝜎 𝑅𝑢𝑡ℎ 𝑑 Ω 𝑐𝑚 = 𝑎 2 4 ∙ 𝑠𝑖𝑛 −4 𝜃 𝑐𝑚 2
~10-22s
𝑑 𝜎 𝑖𝑛𝑒𝑙 𝑑 Ω 𝑐𝑚 = 𝒂 𝒊⟶𝒇 𝟐 ∙ 𝑑 𝜎 𝑅𝑢𝑡ℎ 𝑑 Ω 𝑐𝑚 
lifetime
~ s If
1st order: Ii
𝟏 𝝉 ≅𝟏.𝟐𝟑∙ 𝟏𝟎 𝟏𝟑 ∙𝑩 𝑬𝟐; 𝑰 𝒇 → 𝑰 𝒊 ∙ 𝑬 𝜸 𝟓 𝒔 −𝟏
𝐵 𝐸2; 𝐼 𝑖 → 𝐼 𝑓 = 1 2 𝐼 𝑖 𝐼 𝑓 𝑀 𝐸2 𝐼 𝑖 2
The inelastic cross section
is a direct measure of the E2 matrix elements
𝑑 𝜎 𝑖𝑛𝑒𝑙 𝑑 Ω 𝑐𝑚
H.J. Wollersheim et al., Phys. Lett 48B (1974) 323

4. Coulomb excitation Sommerfeld parameter:
wave
particle
Sommerfeld parameter:
>> 1 requirement for a (semi-) classical treatment
of equations of motion (hyperbolic trajectories )
4He (Z=2) projectiles behave like waves
quantummechanical analysis is needed

5. Classical Coulomb trajectories
impact parameter
scattering angle
Hyperbolic trajectory:
ε = sin-1(θcm/2) eccentricity of orbit
𝑡= 𝑎 𝑣 ∞ 𝜖∙𝑠𝑖𝑛ℎ𝜔+𝜔
distance of closest approach:
impact parameter:
angular momentum :
𝐷=𝑎∙ 𝑠𝑖𝑛 −1 𝜃 𝑐𝑚 2 +1
𝑏=𝑎∙𝑐𝑜𝑡 𝜃 𝑐𝑚 2
ℓ= 𝑘 ∞ ∙𝑏=𝜂∙𝑐𝑜𝑡 𝜃 𝑐𝑚 2

6. Safe bombarding energy pure electromagnetic interaction
Nuclear interaction radius:
CP, CT half-density radii
Pure Coulomb excitation requires a much larger distance between the nuclei
 ”safe bombarding energy” requirement
𝑅 𝑖𝑛𝑡 ≅ 𝐶 𝑝 + 𝐶 𝑡 +3𝑓𝑚
𝑅 𝑖𝑛𝑡 = 𝐶 𝑝 + 𝐶 𝑡 +4.49− 𝐶 𝑝 + 𝐶 𝑡 6.35

7. Safe bombarding energy pure electromagnetic interaction
< 1% deviation from Coulomb excitation
Dmin
Rutherford scattering only if Dmin is large
compared to nuclear radii + surfaces:
Dmin > Cp + Ct + 5 fm
CP, CT half-density radii
choose adequate beam energy Elab
(D > Dmin for all θcm)

8. Electric fields of multipoles
In general the electric potential due to
an arbitrary charge distribution is
𝑈 𝑟 = 𝜌 𝑝 𝑟 ′ 𝑟 − 𝑟 ′ 𝑑𝜏′
expansion
1 𝑟−𝑟′ = ℓ=0 ∞ 𝑟′ ℓ 𝑟 ℓ+1 4𝜋 2ℓ+1 𝑚=−ℓ ℓ 𝑌 ℓ𝑚 𝜗,𝜑 𝑌 ℓ𝑚 ∗ 𝜗 ′ ,𝜑′
ℓ=𝑚= 𝑌 00 𝜗,𝜑 = 𝑌 00 𝜗 ′ ,𝜑′ = 1 4𝜋
special case: electric monopole
→ 𝑟−𝑟′ = 1 𝑟

9. Electric fields of multipoles
In general the electric potential due to
an arbitrary charge distribution is
𝑈 𝑟 = 𝜌 𝑝 𝑟 ′ 𝑟 𝑑𝜏′
homogenous charge distribution
𝜌 𝑝 𝑟′ = 3∙𝑍𝑒 4𝜋∙ 𝑅 0 3
𝑈 𝑟 = 3∙𝑍𝑒 4𝜋∙ 𝑅 𝑟 𝑟′ 2 𝑑 𝑟 ′ 𝑠𝑖𝑛 𝜗 ′ 𝑑 𝜗 ′ 𝑑𝜑′
special case: electric monopole
𝑈 𝑟 = 3∙𝑍𝑒 4𝜋∙ 𝑅 0 3 ∙ 1 𝑟 ∙ 𝑅 ∙4𝜋
= 𝒁𝒆 𝒓

10. Electric fields of multipoles
In general the electric potential due to
an arbitrary charge distribution is
𝑈 𝑟 = 𝜌 𝑝 𝑟 ′ 𝑟 − 𝑟 ′ 𝑑𝜏′
1 𝑟−𝑟′ = 1 𝑟 𝑓𝑜𝑟 𝑟>𝑟′ 1 𝑟′ 𝑓𝑜𝑟 𝑟<𝑟′
𝑈 𝑟 =4𝜋∙ 3∙𝑍𝑒 4𝜋∙ 𝑅 0 3 ∙ 0 𝑟 1 𝑟 ∙ 𝑟′ 2 𝑑𝑟′ + 𝑟 𝑅 𝑟′ ∙ 𝑟′ 2 𝑑𝑟′
𝑈 𝑟 = 3∙𝑍𝑒 𝑅 0 3 ∙ 𝑟 𝑅 − 𝑟 2 2
special case: electric monopole
𝑈 𝑟 = 𝑍𝑒 2 𝑅 0 ∙ 3− 𝑟 𝑅

11. Electric fields of multipoles
In general the electric potential due to
an arbitrary charge distribution is
𝑈 𝑟 = 𝜌 𝑝 𝑟 ′ 𝑟 − 𝑟 ′ 𝑑𝜏′
expansion
1 𝑟−𝑟′ = ℓ=0 ∞ 𝑟′ ℓ 𝑟 ℓ+1 4𝜋 2ℓ+1 𝑚=−ℓ ℓ 𝑌 ℓ𝑚 𝜗,𝜑 𝑌 ℓ𝑚 ∗ 𝜗 ′ ,𝜑′
multipole moments
𝑀 ∗ ℓ,𝑚 = 𝜌 𝑝 𝑟′ ∙ 𝑟′ ℓ ∙ 𝑌 ℓ𝑚 ∗ 𝜗 ′ ,𝜑′ 𝑑𝜏′
𝑀 ∗ ℓ=2,𝑚 = 3∙𝑍𝑒∙ 𝑅 𝜋 ∙ 𝛽 2
special case: electric quadrupole matrix element
𝐵 𝐸ℓ=2; 𝐼 𝑖 → 𝐼 𝑓 = 𝑀 𝑓 𝑚 𝐼 𝑓 𝑀 𝑓 𝐾 𝑓 𝑀 ℓ=2,𝑚 𝐼 𝑖 𝑀 𝑖 𝐾 𝑖 2
B(E2)-value:

12. Electric fields of multipoles
multipole moments
𝑀 ∗ ℓ,𝑚 = 𝜌 𝑝 𝑟′ ∙ 𝑟′ ℓ ∙ 𝑌 ℓ𝑚 ∗ 𝜗 ′ ,𝜑′ 𝑑𝜏′
special case: electric quadrupole matrix element (ℓ = 2, m = 0)
𝜌 𝑝 𝑟′ = 3∙𝑍𝑒 4𝜋∙ 𝑅 0 3
𝑀 ∗ 2,0 = 3∙𝑍𝑒 4𝜋∙ 𝑅 𝑟′ 2 ∙ 𝑌 20 ∗ 𝜗 ′ , 𝜑 ′ ∙ 𝑟′ 2 𝑑 𝑟 ′ 𝑠𝑖𝑛 𝜗 ′ 𝑑 𝜗 ′ 𝑑𝜑′
𝑀 ∗ 2,0 = 3∙𝑍𝑒 4𝜋∙ 𝑅 0 3 ∙ 𝑅 𝛽 2 𝑌 ∙ 𝑌 20 ∗ ∙𝑠𝑖𝑛 𝜗 ′ 𝑑 𝜗 ′ 𝑑𝜑′
𝑅 𝜗 ′ ,𝜑′ = 𝑅 0 ∙ 1+ 𝛽 2 𝑌 20 𝜗 ′ ,𝜑′
𝑀 ∗ 2,0 ≅ 3∙𝑍𝑒∙ 𝑅 𝜋 ∙ 𝛽 2 𝑌 20 ∙ 𝑌 20 ∗ ∙𝑑Ω′
𝑀 ∗ 2,0 ≅ 3∙𝑍𝑒∙ 𝑅 𝜋 ∙ 𝛽 2

13. Electric fields of multipoles
In general the electric potential due to
an arbitrary charge distribution is
𝑈 𝑟 = 𝜌 𝑝 𝑟 ′ 𝑟 − 𝑟 ′ 𝑑𝜏′
expansion
1 𝑟−𝑟′ = ℓ=0 ∞ 𝑟′ ℓ 𝑟 ℓ+1 4𝜋 2ℓ+1 𝑚=−ℓ ℓ 𝑌 ℓ𝑚 𝜗,𝜑 𝑌 ℓ𝑚 ∗ 𝜗 ′ ,𝜑′
multipole moments
𝑀 ∗ ℓ,𝑚 = 𝜌 𝑝 𝑟′ ∙ 𝑟′ ℓ ∙ 𝑌 ℓ𝑚 ∗ 𝜗 ′ ,𝜑′ 𝑑𝜏′
𝑈 𝑟 = 𝑚=−2 𝑚=2 4𝜋 5 ∙ 1 𝑟 3 ∙ 𝑌 ℓ=2,𝑚 𝜗,𝜑 ∙ 𝑀 ∗ ℓ=2,𝑚
special case: electric quadrupole potential

14. Monopole, dipole, quadrupole, ...
octopole
𝑈 𝑟 ∝ 1 𝑟
𝑈 𝑟 ∝ 1 𝑟 2
𝑈 𝑟 ∝ 1 𝑟 3
𝑈 𝑟 ∝ 1 𝑟 4

15. Kinematics
58Ni → 175 MeV Ψ: ϑ:

16. Coulomb excitation at IUAC
58Ni → 122Sn at 175 MeV
Esafe = 202 MeV
beam intensity = 0.5 pnA
target thickness = 0.5 mg/cm2
→ luminosity = 8·1027 s-1cm-2
cross section ~ 70 mb
event rate = 560 s-1
γ-efficiency = 0.005
pγ-rate (Sn) = 3/s
112Sn target
58Ni beam
Clover Ge detector
R. Kumar et al., Phys. Rev. C81, (2010)
M. Saxena et al., Phys. Rev. C90, (2014)

17. Coulomb excitation at IUAC
58Ni → 122Sn at 175 MeV
112Sn target
58Ni beam
Clover Ge detector
R. Kumar et al., Phys. Rev. C81, (2010)
M. Saxena et al., Phys. Rev. C90, (2014)

18. Annular gas-filled parallel-plate avalanche counter (PPAC)
V0 ~ 500 V
p = 5-10 Torr
gap ~ 3 mm (anode-cathode)
122Sn
58Ni beam
Δt ~ tanϑ i o i
delay line
Δtime →
A. Jhingan detector laboratory at IUAC

19. Doppler shift correction
58Ni + 122Sn at 175 MeV
delay line: inner – outer contact ≈ tanϑ
φ-segmentation : 360, 720, 1080, etc
𝑡𝑎𝑛𝜗= 𝑡𝑎𝑛 45 0 −𝑡𝑎𝑛 𝑐ℎ 2 − 𝑐ℎ 1 ∙ 𝑐ℎ− 𝑐ℎ 1 +𝑡𝑎𝑛 15 0
58Ni projectile measured with PPAC (122Sn target excitation)
index 1 ≡ projectile (58Ni) index 2 ≡ target nucleus (122Sn)
𝑣 𝑐𝑚 = ∙ 1+ 𝐴 2 𝐴 1 −1 𝐸 𝑙𝑎𝑏 𝐴 1
( = )
112Sn target
𝜃 𝑐𝑚 = 𝜗 1 +𝑎𝑟𝑐𝑠𝑖𝑛 𝐴 1 𝐴 2 𝑠𝑖𝑛 𝜗 1
𝜗 2 =0.5∙ − 𝜃 𝑐𝑚
𝑣 2 =2∙ 𝑣 𝑐𝑚 ∙𝑐𝑜𝑠 𝜗 2
𝑐𝑜𝑠 𝜗 𝛾2 =𝑐𝑜𝑠 𝜗 𝛾 ∙𝑐𝑜𝑠 𝜗 2 −𝑠𝑖𝑛 𝜗 𝛾 ∙𝑠𝑖𝑛 𝜗 2 ∙𝑐𝑜𝑠 𝜑 𝛾 − 𝜑 2
𝑐𝑜𝑠 𝜑 𝛾 − 𝜑 1 =𝑐𝑜𝑠 𝜑 𝛾 ∙𝑐𝑜𝑠 𝜑 1 +𝑠𝑖𝑛 𝜑 𝛾 ∙𝑠𝑖𝑛 𝜑 1
𝐸 𝛾0 𝐸 𝛾 = 1− 𝑣 2 ∙𝑐𝑜𝑠 𝜗 𝛾2 1− 𝑣 2 2
58Ni beam
D. Schwalm et al. Nucl.Phys. A192 (1972), 449

20. Doppler shift correction
58Ni + 122Sn at 175 MeV
112Sn target
2+ 122Sn
2+ 58Ni
58Ni beam
R. Kumar, A. IUAC

21. Experimental set-up TARGET CLOVER DETECTORS @ backward angles
forward angle

22. Particle-gamma coincidence spectroscopy+ b
projectile
target +
~10-22s 
lifetime
~ s 2+
1st order: 0+
Coulomb excitation:
𝝈 𝑬𝟐 𝟐 + 𝑏 =4.819∙ 1+ 𝐴 1 𝐴 2 −2 ∙ 𝐴 1 𝑍 2 2 ∙ 𝐸 𝑀𝑒𝑉 −∆ 𝐸′ 𝑀𝑒𝑉 ∙ 𝑩 𝑬𝟐, 𝟎 + → 𝟐 + 𝑒 2 𝑏 2 ∙ 𝑓 𝐸2 𝜉
𝜉= 𝑍 1 ∙𝑍 2 ∙ 𝐴 1 1/2 ∙∆ 𝐸′ 𝑀𝑒𝑉 ∙ 𝐸 𝑀𝑒𝑉 −0.5∙∆ 𝐸′ 𝑀𝑒𝑉 3/2 ∙ ∆𝐸′ 𝐸 2 +…

23. Particle-gamma coincidence spectroscopy
Coulomb excitation:
𝝈 𝑬𝟐 𝟐 + 𝑏 =4.819∙ 1+ 𝐴 1 𝐴 2 −2 ∙ 𝐴 1 𝑍 2 2 ∙ 𝐸 𝑀𝑒𝑉 −∆ 𝐸′ 𝑀𝑒𝑉 ∙ 𝑩 𝑬𝟐, 𝟎 + → 𝟐 + 𝑒 2 𝑏 2 ∙ 𝑓 𝐸2 𝜉
𝜉= 𝑍 1 ∙𝑍 2 ∙ 𝐴 1 1/2 ∙∆ 𝐸′ 𝑀𝑒𝑉 ∙ 𝐸 𝑀𝑒𝑉 −0.5∙∆ 𝐸′ 𝑀𝑒𝑉 3/2 ∙ ∆𝐸′ 𝐸 2 +…
K. Alder et al., RMP 28 (56) 432

24. Particle-gamma coincidence spectroscopy+ b
projectile
target +
~10-22s 
lifetime
~ s 2+
1st order: 0+
Coulomb excitation:
𝝈 𝑬𝟐 𝟐 + 𝑏 =4.819∙ 1+ 𝐴 1 𝐴 2 −2 ∙ 𝐴 1 𝑍 2 2 ∙ 𝐸 𝑀𝑒𝑉 −∆ 𝐸′ 𝑀𝑒𝑉 ∙ 𝑩 𝑬𝟐, 𝟎 + → 𝟐 + 𝑒 2 𝑏 2 ∙ 𝑓 𝐸2 𝜉
Particle-γ coincidence spectroscopy: Yγ(Ii→If) → σE2(Ii)
(indirect measurement)
need to take into account branching ratios, particle-γ angular correlations, electron conversion

25. Particle-gamma angular correlation
112Sn
58Ni beam

26. Conversion electrons Energetics of CE-decay (i=K, L, M,….)
Ei = Ef + Ece,i + EBE,i
γ- and CE-decays are independent; transition probability (λ ~ Intensity)
λT = λγ + λCE = λγ + λK + λL + λM……
Conversion coefficient
ai = λCE,i / λγ

27. Coulomb excitation results of semi-magic Sn isotopes
Z = 50
g7/2
d5/2
s1/2
d3/2
h11/2
single-particle transition:
(Weisskopf estimate)
𝐵 𝐸2; 𝐼 𝑖 → 𝐼 𝑔𝑠 = −6 ∙ 𝐴 4/ 𝑒 2 𝑏 2
𝐵 𝐸2; 2 + → 0 + =14 𝑠𝑝𝑢 for 114Sn
𝐵 𝐸2; 0 + → 2 + = 𝑒 2 𝑏 for 114Sn
𝐵 𝐸2; 𝐼 𝑖 → 𝐼 𝑓 = 2 𝐼 𝑓 +1 2 𝐼 𝑖 +1 ∙𝐵 𝐸2; 𝐼 𝑓 → 𝐼 𝑖
N = 50
N = 82

28. Coulomb excitation results of semi-magic Sn isotopes
Z = 50
g7/2
d5/2
s1/2
d3/2
h11/2
seniority scheme

29. Coulomb excitation team
Vivek Mishra, Pieter Doornenbal, Mansi Saxena, Chhavi Joshi, Sunil Prajapati, Rakesh Kumar, Paer-Anders Soederstroem, Mohit Kumar, Sunil Dutt, Akhil Jhingan, Aakashrup Banerjee, Hans Jürgen Wollersheim.

30. Multiple (multi-step) Coulomb excitation

31. γ-ray decal after multiple Coulomb excitation

32. The reorientation effect
The excitation cross section is a
direct measure of the E matrix elements.
reorientation effect: If Ii Mf If
1st order: Ii 
𝑃 0→2 (2) 𝜃,𝜉 =𝑃 0→2 (1) 𝜃,𝜉 ∙ 𝜋 ∙ 𝐴 𝑝 𝑍 𝑝 ∙ ∆𝐸 1+ 𝐴 𝑝 𝐴 𝑡 ∙ 𝑄 2 ∙𝐾 𝜃,𝜉
𝑄 =− 2𝜋 ∙ 2 𝑀 𝐸2 2

33. Shape coexistence in 74Kr

34. Deformed nuclei z b a 𝑅 𝜃,𝜙 = 𝑅 0 ∙ 1+𝛽∙ 𝑌 20 𝜃,𝜙 collective rotation
𝛽= 𝜋 5 Δ𝑅 𝑅
𝑅 = 𝑎+𝑏 2
𝑅 𝜃,𝜙 = 𝑅 0 ∙ 1+𝛽∙ 𝑌 20 𝜃,𝜙
Δ𝑅=𝑎−𝑏
𝐸 𝛾 = 𝐸 𝐼 − 𝐸 𝐼−2 = ℏ 2 2ℑ 4𝐼−2
𝐵 𝐸2;𝐼→𝐼−2 = 15 32𝜋 𝐼 𝐼+1 2𝐼−1 2𝐼+1 ∙ 𝑄 2
𝐸 𝐼 = ℏ 2 2ℑ 𝐼 𝐼+1
𝑄 2 = 3𝑍 𝑅 𝜋 ∙𝛽
ℑ= 2 5 𝐴∙𝑀∙ 𝑅 0 2 ∙ 𝛽 2
analysis with GOSIA code
W. Spreng et al., Phys. Rev. Lett. 51 (1983), 1522

35. angular momentum transfer
Coulomb excitation
angular momentum transfer
164Dy(208Pb, 208Pb´)
EPb = 4.7 MeV/A
4+→2+
6+→4+
8+→6+
10+→8+
12+→10+
θcm=
θcm=
θcm=
14+→12+
16+→14+
18+→16+
∆ 𝐿 𝑚𝑎𝑥 = 3 2 ∙ 𝐽 20 𝜃 𝑐𝑚 ∙𝑞
𝑞= 𝑍 𝑝 ∙ 𝑒 2 ∙ 𝑄 0 4∙ℏ∙𝑣∙ 𝑎 2
𝐽 20 𝜃 𝑐𝑚 = 𝑠𝑖𝑛 2 𝜃 𝑐𝑚 2 + 𝑡𝑎𝑛 2 𝜃 𝑐𝑚 2 1− 𝜋− 𝜃 𝑐𝑚 2 𝑡𝑎𝑛 𝜃 𝑐𝑚 2
𝐽 20 𝜃 𝑐𝑚 ≅ −𝑐𝑜𝑠 𝜃 𝑐𝑚

36. Coulomb excitation 𝜉=1 energy transfer 𝜉 𝜃 𝑐𝑚 = ∆ 𝐸 𝑒𝑥𝑐 ℏ∙𝑐 ∙ 𝐷−𝑎 𝛾∙𝛽
𝜉 𝜃 𝑐𝑚 = ∆ 𝐸 𝑒𝑥𝑐 ℏ∙𝑐 ∙ 𝐷−𝑎 𝛾∙𝛽
ξ measures suddenness of interaction
“adiabatic limit” for (single step) excitation ξ = 1 VC
∆ 𝐸 𝑒𝑥𝑐 =ℏ∙𝑐∙ 𝛽∙𝛾 𝐷−𝑎
maximum energy transfer:

37. Evolution of nuclear structure as a function of nucleon number