Download Elastic Scattering

Elastic Scattering
Hans-Jürgen Wollersheim
Classification of heavy ion collisions









elastic scattering
Fresnel & Fraunhofer scattering
scattering parameters
differential cross section
optical model analysis
nuclear radius
total reaction cross section
cross sections at high energy
influence of nuclear structure
partial cross section vs. angular momentum
elastic scattering
b
elastic scattering
b
elastic scattering
b
elastic scattering
b
nucleon transfer
elastic scattering
b≈0
nucleon transfer
elastic scattering
b≈0
nucleon transfer
elastic scattering
b≈0
compound nucleus
formation
nucleon transfer
Nuclear reaction cross sections
Consider a beam of projectiles of intensity Φa particles/sec which hits a thin foil of
target nuclei with the result that the beam is attenuated by reactions in the foil such
that the transmitted intensity is Φ particles/sec.
The fraction of the incident particles disappear from the beam, i.e. react, in passing
through the foil is given by
A
d
va
𝑑Φ = −Φ ∙ 𝑛𝑏 ∙ 𝜎 ∙ 𝑑𝑥
The number of reactions that are occurring is the difference between the initial and
transmitted flux
Φ𝑖𝑛𝑖𝑡𝑖𝑎𝑙 − Φ𝑡𝑟𝑎𝑛𝑠 = Φ𝑖𝑛𝑖𝑡𝑖𝑎𝑙 1 − 𝑒𝑥𝑝 −𝑛𝑏 ∙ 𝑑 ∙ 𝜎
≈ Φ𝑖𝑛𝑖𝑡𝑖𝑎𝑙 ∙ 𝑁𝑏 ∙ 𝜎
(for thin target)
Φa = na·va
Example:
A particle current of 1 pnA consists of 6·109 projectiles/s.
A 132Sn target (1 mg/cm2) consists of 5·1018 nuclei/cm2
6 ∙ 1023 ∙ 10−3 𝑔 𝑐𝑚2
= 4.5 ∙ 1018
132𝑔
𝑡𝑎𝑟𝑔𝑒𝑡 𝑛𝑢𝑐𝑙𝑒𝑖
𝑐𝑚2
Luminosity = projectiles [s-1] · target nuclei [cm-2]
Luminosity (projectile → 132Sn) = 3·1028 [s-1cm-2]
Reaction rate [s-1] = luminosity · cross section [cm2]
= projectiles [s-1] · target nuclei [cm-2] · cross section [cm2]
Nb = nb·A·d
Elastic Scattering
Fraunhofer (left) and Fresnel (right) diffraction
12C
+ 16O
16O
η=2
+ 208Pb
η = 32
Born approximation (quantum description) or classical description:
half distance of closest approach for head-on collision
wave length of projectile
ƛ = (k∞)-1
0.72 ∙ 𝑍1 𝑍2 𝐴1 + 𝐴2
∙
𝑓𝑚
𝑇𝑙𝑎𝑏
𝐴2
𝐴2
= 0.219 ∙
∙ 𝐴1 ∙ 𝑇𝑙𝑎𝑏
𝐴1 + 𝐴2
𝑎
𝜆ƛ
𝜂 = 𝑘∞ ∙ 𝑎 = 0.157 ∙ 𝑍1 𝑍2 ∙
𝑎=
𝑘∞
𝜂=
𝑓𝑚−1
𝐴1
𝑇𝑙𝑎𝑏
Elastic Scattering
4He
+ 209Bi
Elab = 69.5 MeV
4He
+ 209Bi
Elab = 12 MeV
Rutherford scattering
η = 15
4He
+ 209Bi
Elab = 22 MeV
Fresnel scattering
η = 11
Fraunhofer scattering
η=6
Transition from classical (optical) picture to quantum picture
Elastic Scattering
6Li
elastic scattering @ 88 MeV
particle
η = 2.5
η = 10
η = 1.5
η = 3.3
Fraunhofer scattering (η<10)
wave
Oscillation in angular distribution → good angular resolution required
S. Hossain et al. Phys. Scr. 87 (2013) 015201
Fresnel scattering (η≥10)
Scattering parameters
D
impact parameter:
𝜃𝑐𝑚
𝑏 = 𝑎 ∙ 𝑐𝑜𝑡
2
distance of closest approach:
θcm
b
D
center of mass system
𝜃𝑐𝑚
𝐷 = 𝑎 ∙ 𝑠𝑖𝑛−1
+1
2
orbital angular momentum:
𝜃𝑐𝑚
ℓ = 𝑘∞ ∙ 𝑏 = 𝜂 ∙ 𝑐𝑜𝑡
2
half distance of closest approach
in a head-on collision (θcm=1800):
𝑎=
0.72 ∙ 𝑍1 𝑍2 𝐴1 + 𝐴2
∙
𝑇𝑙𝑎𝑏
𝐴2
𝑓𝑚
asymptotic wave number:
𝐴2
𝑘∞ = 0.219 ∙
∙ 𝐴1 ∙ 𝑇𝑙𝑎𝑏
𝐴1 + 𝐴2
Sommerfeld parameter:
𝜂 = 𝑘∞ ∙ 𝑎 = 0.157 ∙ 𝑍1 𝑍2 ∙
𝐴1
𝑇𝑙𝑎𝑏
𝑓𝑚−1
Scattering theory
 Particles from the ring defined by the impact parameter b and b+db scatter between angle θ and θ+dθ
𝑗 ∙ 2𝜋 ∙ 𝑏 ∙ 𝑑𝑏 = 𝑗 ∙ 2𝜋 ∙ 𝑠𝑖𝑛𝜃 ∙
𝑑𝜎
𝑑Ω
𝑑𝜎
𝑏 𝑑𝑏
=
𝑑Ω 𝑠𝑖𝑛𝜃 𝑑𝜃
impact parameter: 𝑏 = 𝑎 ∙ 𝑐𝑜𝑡
ring area: 2πb db
solid angle: 2π sinθ dθ
𝜃
2
𝜃
𝜃
𝜃
𝜃
𝑑𝑏
𝑎 −𝑠𝑖𝑛 2 ∙ 𝑠𝑖𝑛 2 − 𝑐𝑜𝑠 2 ∙ 𝑐𝑜𝑠 2 𝑎
1
= ∙
= ∙
𝜃
𝑑𝜃
2
2 𝑠𝑖𝑛 𝜃
𝑠𝑖𝑛2
2
2
𝜃
𝑐𝑜𝑠
𝑑𝜎
1
𝑎
2∙
=𝑎∙
∙
𝜃
𝜃
𝜃
𝜃
𝑑Ω
𝑠𝑖𝑛
2 ∙ 𝑐𝑜𝑠 ∙ 𝑠𝑖𝑛 2 ∙ 𝑠𝑖𝑛2
2
2
2
2
𝑑𝜎 𝑎2
𝜃
=
∙ 𝑠𝑖𝑛−4
𝑑Ω
4
2
Scattering theory
ring area: 2πb db
solid angle: 2π sinθ dθ
𝑑𝜎 𝑎2
𝜃
−4
=
∙ 𝑠𝑖𝑛
𝑑Ω
4
2
Scattering theory
angular momentum and scattering angle:
ℓ=𝑏∙𝑝=𝑏∙ 2∙𝑚∙𝑇
ℓ = 𝜂 ∙ 𝑐𝑜𝑡
𝜃
2
𝑘∞ =
2∙𝑚∙𝑇
ℏ
𝜂 = 𝑘∞ ∙ 𝑎
𝑑𝜎 𝑑𝜎 𝑑Ω 𝑎2
𝜃 𝑑Ω
=
∙
=
∙ 𝑠𝑖𝑛−4 ∙
𝑑ℓ 𝑑Ω 𝑑ℓ
4
2 𝑑ℓ
𝑑Ω
𝑑𝜃
= 2𝜋 ∙ 𝑠𝑖𝑛𝜃 ∙
𝑑ℓ
𝑑ℓ
2𝜃
𝑑Ω
𝜃
𝜃 2 ∙ 𝑠𝑖𝑛 2
= 2𝜋 ∙ 2 ∙ 𝑐𝑜𝑠 ∙ 𝑠𝑖𝑛 ∙
𝑑ℓ
2
2
𝜂
𝑑𝜎 𝑎2
𝜃 8𝜋 ∙ ℓ
𝜃
=
∙ 𝑠𝑖𝑛−4 ∙ 2 ∙ 𝑠𝑖𝑛4
𝑑ℓ
4
2 𝜂
2
𝑑𝜎 2𝜋
= 2 ∙ℓ
𝑑ℓ 𝑘∞
𝑑ℓ 𝜂
𝜃
= ∙ 𝑠𝑖𝑛−2
𝑑𝜃 2
2
𝑐𝑜𝑠
𝜃 ℓ
𝜃
= ∙ 𝑠𝑖𝑛
2 𝜂
2
Scattering theory
distance of closest approach and scattering angle:
𝐷 = 𝑎 ∙ 𝑠𝑖𝑛−1
𝜃
+1
2
𝑠𝑖𝑛
𝜃
𝑎
=
2 𝐷−𝑎
𝑑𝜎 𝑑𝜎 𝑑Ω 𝑎2
𝜃 𝑑Ω
=
∙
=
∙ 𝑠𝑖𝑛−4 ∙
𝑑𝐷 𝑑Ω 𝑑𝐷
4
2 𝑑𝐷
𝑑Ω
𝑑𝜃
= 2𝜋 ∙ 𝑠𝑖𝑛𝜃 ∙
𝑑𝐷
𝑑𝐷
2𝜃
𝑑Ω
𝜃
𝜃 2 𝑠𝑖𝑛 2
= 2𝜋 ∙ 2 ∙ 𝑐𝑜𝑠 ∙ 𝑠𝑖𝑛 ∙ ∙
𝑑𝐷
2
2 𝑎 𝑐𝑜𝑠 𝜃
2
𝑑Ω 8𝜋
𝜃
=
∙ 𝑠𝑖𝑛3
𝑑𝐷
𝑎
2
𝑑𝜎 𝑎2
𝜃 8𝜋
𝜃 2𝜋 ∙ 𝑎
=
∙ 𝑠𝑖𝑛−4 ∙
∙ 𝑠𝑖𝑛3 =
𝑑𝐷
4
2 𝑎
2 𝑠𝑖𝑛 𝜃
2
𝑑𝜎
= 2𝜋 ∙ 𝐷 − 𝑎
𝑑𝐷
𝜃
𝑑𝐷 𝑎 −𝑐𝑜𝑠 2
= ∙
𝑑𝜃 2 𝑠𝑖𝑛2 𝜃
2
Summary
 impact parameter and scattering angle:
𝜃
𝑍𝑝 ∙ 𝑍𝑡 ∙ 𝑒 2
𝑏 = 𝑎 ∙ 𝑐𝑜𝑡
𝑎=
2
2 ∙ 𝐸𝑐𝑚
𝑑𝜎 𝑎2
𝜃
=
∙ 𝑠𝑖𝑛−4
𝑑Ω
4
2
 angular momentum and scattering angle:
𝜃
𝜂 = 𝑘∞ ∙ 𝑎
ℓ = 𝜂 ∙ 𝑐𝑜𝑡
2
𝑘∞ =
𝑑𝜎 2𝜋
= 2 ∙ℓ
𝑑ℓ 𝑘∞
 distance of closest approach and scattering angle:
𝐷 = 𝑎 ∙ 𝑠𝑖𝑛−1
𝜃
+1
2
𝑑𝜎
= 2𝜋 ∙ 𝐷 − 𝑎
𝑑𝐷
2 ∙ 𝑚 ∙ 𝐸𝑐𝑚
ℏ
Summary
 impact parameter and scattering angle:
𝜃
𝑏 = 𝑎 ∙ 𝑐𝑜𝑡
2
𝑑𝜎 𝑎2
𝜃
=
∙ 𝑠𝑖𝑛−4
𝑑Ω
4
2
 angular momentum and scattering angle:
𝜃
ℓ = 𝜂 ∙ 𝑐𝑜𝑡
2
𝑑𝜎 2𝜋
= 2 ∙ℓ
𝑑ℓ 𝑘∞
 distance of closest approach and scattering angle:
𝐷 = 𝑎 ∙ 𝑠𝑖𝑛−1
𝜃
+1
2
𝑑𝜎
= 2𝜋 ∙ 𝐷 − 𝑎
𝑑𝐷
Nuclear Reactions
elastic scattering deviates from Rutherford scattering
scattering angle
angular momentum
distance of closest approach
Ratio of the elastic scattering and Rutherford scattering (nuclear reactions) is only independent
of the bombarding energy when plotted versus the distance of closest approach D.
Data are from Coulomb excitation experiments: The excitation probability P8(exp) of the low-lying rotational state Iπ = 8+ is not only excited
directly but also fed from higher-lying states and is therefore a measure of the elastic scattering. When comparing with Coulomb excitation
calculations P8(theo), which corresponds to the Rutherford scattering, the observed deviations are a clear indication of nuclear interactions
(nuclear reactions).
Heavy-ion elastic scattering
energy and angular distributions
determine elastic energy = f(θ), fit standard line shape,
determine elastic cross section σel(θ)
plot ratio elastic/Rutherford cross section = f(θ),
determine quarter-point θ1/4
→ total integrated reaction cross section σR
J.R. Birkelund et al., Phys.Rev.C13 (1976), 133
(600 MeV) + 209Bi: θ = 66.70
→ ℓgr = 268 ħ, Rint = 14.2 fm, σR = 1.9 b
84Kr
Heavy-ion elastic scattering and the optical model
V
𝑽𝑪 𝒓 =
 1cm/ 4
𝑽𝒏𝒖𝒄𝒍
𝒁𝟏 𝒁𝟐 𝒆𝟐
𝒓𝟐
𝟑− 𝟐
𝟐 ∙ 𝑹𝑪
𝑹𝑪
𝒁𝟏 𝒁𝟐 𝒆𝟐
𝒓
−𝑽𝟎
𝒓 =
𝒓−𝒓
𝟏 + 𝒆𝒙𝒑 𝒂 𝑹
𝑹
𝑽𝒊𝒎𝒂𝒈 𝒓 =
J.R. Birkelund et al., Phys.Rev.C13 (1976), 133
−𝑾𝟎
𝒓−𝒓
𝟏 + 𝒆𝒙𝒑 𝒂 𝑰
𝑰
𝒓 < 𝑹𝑪
𝒓 ≥ 𝑹𝑪
r
r
Heavy-ion elastic scattering and the optical model
V
𝑽𝑪 𝒓 =
 1cm/ 4
𝑽𝒏𝒖𝒄𝒍
𝒁𝟏 𝒁𝟐 𝒆𝟐
𝒓𝟐
𝟑− 𝟐
𝟐 ∙ 𝑹𝑪
𝑹𝑪
𝒁𝟏 𝒁𝟐 𝒆𝟐
𝒓
−𝑽𝟎
𝒓 =
𝒓−𝒓
𝟏 + 𝒆𝒙𝒑 𝒂 𝑹
𝑹
𝑽𝒊𝒎𝒂𝒈 𝒓 =
𝒓 < 𝑹𝑪
𝒓 ≥ 𝑹𝑪
r
−𝑾𝟎
𝒓−𝒓
𝟏 + 𝒆𝒙𝒑 𝒂 𝑰
𝑰
Parameters of the optical model fit:
θ1/4 = 600 → Rint = 13.4 [fm]
→ ℓgr = 152 [ħ]
J.R. Birkelund et al., Phys.Rev.C13 (1976), 133
V0 (MeV)
rR (fm)
aR (fm)
W0 (MeV)
rI (fm)
aI (fm)
ℓ (ħ)
σR (mb)
68.0
1.167
0.540
83.9
1.167
0.540
150
1937
214.5
1.104
0.536
261.1
1.104
0.536
149
1902
43.2
1.196
0.529
56.0
1.196
0.529
150
1939
Elastic scattering and nuclear radius
Nuclear interaction radius: (distance of closest approach)
𝑅𝑖𝑛𝑡 = 𝐷 = 𝑎 ∙ 𝑠𝑖𝑛−1

cm
1/ 4
𝜃1/4
+1
2
𝑅𝑖𝑛𝑡 = 𝐶𝑝 + 𝐶𝑡 + 4.49 −
𝐶𝑖 = 𝑅𝑖 ∙ 1 − 𝑅𝑖−2
𝑓𝑚
𝐶𝑝 + 𝐶𝑡
6.35
1/3
𝑅𝑖 = 1.28 ∙ 𝐴𝑖
𝑓𝑚
−1/3
− 0.76 + 0.8 ∙ 𝐴𝑖
Nuclear density distributions at the nuclear interaction radius
θ1/4 = 600 → Rint = 13.4 [fm]
→ ℓgr = 152 [ħ]
J.R. Birkelund et al., Phys.Rev.C13 (1976), 133
𝑓𝑚
Nuclear radius
ρ/ρ0
1.0
0.9
0.5
ρ/ρ0
C
0.1
R
nuclear radius of a homogenous charge distribution:
1/3
𝑅𝑖 = 1.28 ∙ 𝐴𝑖
−1/3
− 0.76 + 0.8 ∙ 𝐴𝑖
nuclear radius of a Fermi charge distribution:
r
𝐶𝑖 = 𝑅𝑖 ∙ 1 − 𝑅𝑖−2
𝑓𝑚
𝑓𝑚
Elastic scattering – nuclear reactions
 impact parameter and scattering angle:
𝜃
𝑏 = 𝑎 ∙ 𝑐𝑜𝑡
2
𝑑𝜎 𝑎2
𝜃
=
∙ 𝑠𝑖𝑛−4
𝑑Ω
4
2
θ1/4 = 1320
 angular momentum and scattering angle:
𝜃
ℓ = 𝜂 ∙ 𝑐𝑜𝑡
2
𝑑𝜎 2𝜋
= 2 ∙ℓ
𝑑ℓ 𝑘∞
 distance of closest approach and scattering angle:
𝐷 = 𝑎 ∙ 𝑠𝑖𝑛−1
ℓgr = 206 ħ
𝜃
+1
2
𝑑𝜎
= 2𝜋 ∙ 𝐷 − 𝑎
𝑑𝐷
Rint = 16.2 fm
Total reaction cross section
 impact parameter and scattering angle:
𝜃
𝑏 = 𝑎 ∙ 𝑐𝑜𝑡
2
𝜎𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 = 2𝜋𝑎2 ∙
𝑐𝑚
1 − 𝑐𝑜𝑠 𝜃1/4
−1
− 0.5
θ1/4 = 1320
a = 7.73 fm
ℓgr = 206 ħ
k∞ = 59.9 fm-1
Rint = 16.2 fm
VC(Rint) = 656 MeV
 angular momentum and scattering angle:
𝜃
ℓ = 𝜂 ∙ 𝑐𝑜𝑡
2
𝜎𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 =
𝜋
2 ∙ ℓ𝑔𝑟 ℓ𝑔𝑟 + 1
𝑘∞
 distance of closest approach and scattering angle:
𝐷 = 𝑎 ∙ 𝑠𝑖𝑛−1
𝜃
+1
2
2
𝜎𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 = 𝜋 ∙ 𝑅𝑖𝑛𝑡
∙ 1−
𝑉𝐶 𝑅𝑖𝑛𝑡
𝐸𝑐𝑚
Scattering theory – nuclear absorption
elastic cross section:
𝑑𝜎𝑒𝑙
𝑑𝜎𝑅𝑢𝑡ℎ
= 1 − 𝑃𝑎𝑏𝑠 𝐷
𝑑𝐷
𝑑𝐷
attenuation coefficient:
1 − 𝑃𝑎𝑏𝑠 𝐷
2
= 𝑒𝑥𝑝 −
ℏ
+∞
𝑊 𝑟 𝑡 𝑑𝑡
−∞
proximity potential:
𝑊𝑟 𝑡
= 𝑊0 ∙ 𝑒𝑥𝑝 −
𝑟 𝑡 − 𝐶1 − 𝐶2
𝑎𝐼
attenuation coefficient:
1 − 𝑃𝑎𝑏𝑠 𝐷
2
𝐷 − 𝐶1 − 𝐶2 𝐷
= 𝑒𝑥𝑝 − ∙ 𝑊0 ∙ 𝑒𝑥𝑝 −
∙
ℏ
𝑎𝐼
𝑣
High-energy Coulomb excitation
grazing angle
136Xe
on 208Pb at 700 MeV/u
excitation of giant dipole resonance
Coulomb ex.
Rint  15.0 fm  1/ 4  5.7 mrad
Protons
For relativistic projectiles (  cm  lab):
π
ν
A.Grünschloß et al., Phys. Rev. C60 051601 (1999)
2  Z P ZT e 2 1
D

m0 c 2  2 
Neutrons
High-energy Coulomb excitation
grazing angle
136Xe
on 208Pb at 700 MeV/u
excitation of giant dipole resonance
Coulomb ex.
Rint  15.0 fm  1/ 4  5.7 mrad
Protons
For relativistic projectiles (  cm  lab):
π
ν
A.Grünschloß et al., Phys. Rev. C60 051601 (1999)
2  Z P ZT e 2 1
D

m0 c 2  2 
Neutrons
Effect of nuclear structure on elastic scattering
9,10Be+64Zn
elastic scattering angular distributions @ 29MeV
9Be
Sn=1.665 MeV
10Be
Sn=6.88 MeV
A. Di Pietro et al. Phys. Rev. Lett. 105, 022701(2010)
Effect of nuclear structure on elastic scattering
11Be
10,11Be+64Zn
@ Rex-Isolde, CERN
10Be+64Zn
11Be+64Zn
Optical Model analysis:
 Volume potential responsible for the
core-target interaction obtained from
the 10Be+64Zn elastic scattering fit.
 plus a complex surface DPP having
the shape of a W-S derivative with a
very large diffuseness.
 Very large diffuseness: ai = 3.5 fm
similar to what found in
A. Bonaccrso, NP A706 (2002), 322
Reaction cross-sections
sR(9Be) ≈ 1.1b sR(10Be) ≈ 1.2b sR(11Be) ≈ 2.7b
A. Di Pietro et al. Phys. Rev. Lett. 105, 022701(2010)