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1
Master NPAC:
An introduction to the theory of nuclear
reactions
Guillaume Hupin
IPN Orsayprepared with inputs of D. Lacroix
Lecture 1 : Generalities from classical to
quantum scattering
22
Generalities from classical to quantum scattering
Two-body quantum scattering: phase-shifts, resonances…
Formal theory of scattering to optical potential Gen
era
l asp
ects
Inelastic channels, Fusion
Direct reactions: continuum effect, Break-up, knock-out
Nu
cle
ar
Ph
ysic
s
Microscopic approaches to reactions: TDHF
Few-body approaches to reactions
Researc
h t
op
ics
33
Quantum Collision Theory C. J. Joachain, ISBN 978-0-
444-86773-5, North-Holland, 1975.
Scattering Theory, J. R. Taylor, 978-0-486-45013-1,
Dover publications, 1983.
Introduction to Nuclear Reactions, C. A. Bertulani and
P. Danielewicz,978-0-750-30932-5, Taylor & Francis,
2003.
Nuclear Reactions for Astrophysics, I. J. Thompson and
F. M. Nunes, 978-0-524-85635-5, Cambridge University
press, 2009.
Nuclear Reactions, H.P. gen. Schieck, 978-3-642-
53985-5, Springer, 2014.
Introduction to Nuclear Reactions, G.R. Satchler, 0-
333-25907-6, Macmillan Press, 1980.
44
IPNO
P. Napolitani
G. Hupin
D. Lacroix
J. Carbonell
CEA
M. Dupuis
G. Blanchon
U. of Surrey
P. Stevenson
N. Timofeyuk
A. Diaz-Torres
GANIL
M. Płoszajczak
IPHC
R. Lazauskas
M. Dufour
U. of Varsaw
P. Magierski
U. of Sevilla
A. Moro
J. Lay Valera
ULB
D. Baye
P. Descouvemont
P. Capel
U. of Pisa
A. Kievsky
L. Marcucci
M. Viviani
A. Bonaccorso
GSI
S. Typel
T. Neff
U. of Bochum
E. Epelbaum
U. of Erlangen
P.-G Reinhardt
CSIC
E. Garrido
U. of Trento
G. Orlandini
W. Leigmann
U. Jagiellonian
R. Skibiń
J. Golak
H. Witała
U. of Vilnius
A. Deltuva
Jülich
A. Nogga
J. Haidenbauer
U.-G. Meißner
U. of Lisboa
A. Fonseca
Missing:
Nuclear data evaluators
55
High energy picture: from CMS (CERN)
The low energy nuclear physicist viewpoint
• NN-interaction is a fundamental force
• Nucleons are considered as
elementary (point-like) particles
Here we consider that there is not enough
energy to excite internal degrees of freedom of
nucleons
From D. Furnstahl, Nucl. Phys. B Proc. Suppl. (2012).
~ hard scale
Beam energy < 140 MeV/nucleon (𝜋 threshold)
p p
p
p
66
Typical experiment:
𝑒−, 𝛾, p, n, t, 3He…Heavy Ions…
Beam control:
Beam energy, reactant polarization
Target choice
𝑒−, 𝛾, p, n, t, 3He…Heavy
Ions…polarimeter,
atomic transitions…
Role of reaction theory:
Reconstruct the unseen
history of the reaction
Expt.
parameters
Detection
Theory
77
Nowadays, scientists are
interested in Radioactive
Ion Beams (RIB) to
probe the physics far
from the stability/ of
extreme isospin/ of
astrophysical interest/ …
88
An example of experiment
• Precisely measure the ejectile
• Particle ID with good energy and angles
• measurement
Agata:
4𝜋 gamma-ray detector
FWHM 6keV @ 1 MeV
~43%VAMOS spectrometer
99
Reactions observables depends on the wavelength of the
projectile and how it interacts with target
Why do we need so many beams/detectors ?
Nucleon-nucleon collisions
or
1H
p n
1S0
NN interactions
Charge densities𝑒−Electron scattering
𝑒−
1010
Knockout reactions (high energy) Spectroscopic information
𝑡+∞
𝑡−∞
Transfer reactions (low energy) Spectroscopic information
𝑡−∞
𝑡+∞
Some reactions are designed to study nuclear structure
1111
Coulomb excitation Collective motion
𝛾
Others give access to specific mode of excitations
Photonuclear reactions
GDR excitation𝛾
Also Bremsstrahlung,
radiative capture𝛾 𝑡−∞ 𝑡+∞
1212
Out of equilibrium properties (away from nuclear density + cold)
Reactions at Fermi energy
(30 to 150 MeV/A)
Fragmentation
𝑡−∞ 𝑡+∞
1313
Spectator/participant
fragmentation
Fusion
evaporation
Fusion
fission Vaporization
Quasi-fission
TransferQuasi-elastic
Beam EnergyFusion barrier (5-30 MeV/A) Fermi-Energy (30-100 MeV/A)
Typical excitation energy
8-10 MeV/A3-10 MeV/A0-3 MeV/A
Fragmentation
Incoming channel
𝑡−∞𝑡+∞
We control
Formation of a
compound nucleus
Thermalization
Dinucleus
𝜸𝜸𝜸
1414
Mass/Charge:
• Projectile 𝑁𝑝, 𝑍𝑝• Target 𝑁𝑡, 𝑍𝑡
Beam Energy
• ൗ𝐸𝑏𝐴 , 𝐸𝑝
Spin moments
• Projectile 𝑝𝑧, 𝑝𝑧𝑧…• Target 𝑞𝑧 , 𝑞𝑧𝑧 …
How the probe interacts
• Strong nuclear
• Electromagnetic
• Weak
1515
????
Inclusive/exclusive
Strongly detectors
dependent
Exit channel
Means that X is not in its ground state
Entrance channel
b
Ep
𝑅𝑐.𝑚.
𝐴𝑃, 𝑍𝑃
𝐴𝑇 , 𝑍𝑇Ԧ𝑆
Ԧ𝑆
Elastic scattering 𝐴(𝑎, 𝑎)𝐴
Inelastic scattering
Deeply inelastic…
Compound nucleus
𝐴 + 𝑎՜𝐴 + 𝑎
𝐴 + 𝑎՜𝐵 + 𝑏
𝐴 + 𝑎՜𝐵∗ + 𝑏∗ +⋯
𝐴 + Ԧ𝑎՜ Ԧ𝐴 + 𝑎 Spin transfer 𝐴( Ԧ𝑎, 𝑎) Ԧ𝐴
𝐴 + 𝑎՜𝐶∗ ՜ decay
Transfer reaction 𝐴 𝑎, 𝑏 𝐵
𝐴 + 𝑎՜𝐴 + 𝑎 + 𝛾 Nuclear Bremsstrahlung
No e
nerg
y
tran
sfe
r
En
erg
y
tran
sfe
r
1616
Total charge:
𝑖
𝑍𝑖 =𝑓
𝑍𝑓
Total Baryonic number; i.e. when 𝐸 < 𝐸𝜋 treshold protons and neutron
are equilibrated:
𝑖
𝑍𝑖 =𝑓
𝑍𝑓
𝑖
𝑁𝑖 =𝑓
𝑁𝑓
Total energy
Total linear momentum
Total angular momentum
Total parity
Total isospin
Important kinematics relations
through, which one related quantities
to initial parameter
𝐽𝑖𝜋𝑖𝑇𝑖 = 𝐽𝑓
𝜋𝑓𝑇𝑓where
𝐴 𝐼𝐴 , 𝜋𝐴, 𝑇𝐴 , 𝑎 , 𝐼𝑎𝜋𝑎, 𝑇𝑎𝑆𝜓 𝑟
𝐽𝑖𝜋𝑖𝑇𝑖
𝐵 𝐼𝐵 , 𝜋𝐵, 𝑇𝐵 , 𝑏 𝐼𝑏𝜋𝑏 , 𝑇𝑏𝑆 ෨𝜓 𝑟
𝐽𝑓𝜋𝑓𝑇𝑓
1717
Challenge of nuclear reaction theory
Understand the reaction mechanism to get information on nuclei
Explain the diversity of nuclear phenomena
Give a unified description of these phenomena
Challenge of this lecture
Gives some hint on how these (some) phenomena are described
Gives you some starting point/minimal background on nuclear
reactions models
Gives you overview of hot topics in the field
18
Scattering in Classical
Mechanics
As an introduction
1919
Total cross-section
(Number of scattered particle per unit time)(𝜃, 𝜑)
(Incident flux) (Scattering centers, n)(unit of solid angle)
Differential cross-section
𝑑𝜎channel 𝜃,𝜑
𝑑Ω=
𝑛=1 𝑗𝑓 𝜃, 𝜑
𝑗𝑖
𝜎tot = න
c
𝑑𝜎c 𝜃, 𝜑
𝑑Ω𝑑Ω
2020
In classical mechanic, equations are deterministic i.e. the trajectory depends on b
(or equivalently 𝑏 𝜋 ).
So number of scattered particle per unit time:
𝑗𝑖𝜋 𝑏 + 𝑑𝑏 2 − 𝑏2 ~𝑗𝑖2𝜋𝑏𝑑𝑏
The differential cross-section is:
𝑑𝜎c 𝜃, 𝜑
𝑑Ω=
𝑏
sin 𝜃
𝑑𝑏
𝑑𝜃
2121
2222
2323
Geiger-Marsden experiment (1908-1913)
E < 7.7 MeV
Data: http://hyperphysics.phy-astr.gsu.edu
𝑑𝜎
𝑑𝛺=
𝑍1𝑍2𝑒2
8𝜋𝜀0𝑚𝑣02
2
csc4𝜃
2
2424
• Detector at a fixed 𝜃 angle
• Scan for various beam energy
Eisberg, R. M. and Porter, C. E., Rev. Mod. Phys. 33, 190 (1961)
Geiger-Marsden
data
Rutherford
formula
Exp.
𝑑𝜎
𝑑𝛺=
𝑍1𝑍2𝑒2
8𝜋𝜀0𝑚𝑣02
2
csc4𝜃
2
2525
Eisberg, R. M. and Porter, C. E., Rev. Mod. Phys. 33, 190 (1961)
Rutherford
formula
Exp.
Alpha particle are not
point like-particle
But this alone cannot explain the deviations
from Rutherford
• Nucleons do interact also through
the nuclear force.
• This interaction is a short-range
interaction
grazing
𝑑0
𝐸
𝑑0 =𝑒2𝑍1𝑍22𝜋𝜀02𝐸
2626
Nucleons or 𝛼 scattering on nuclei are
sensitive to both the nucleus spatial
extension and the nuclear interaction
Electron scattering is mostly sensitive to
charged matter (proton) densitySee Z protons
To probe nuclear properties with electrons, high energy electron beam are needed
(𝑚𝑒 ≪ 𝑚𝛼)
𝛽 =𝑣
𝑐is not small and relativistic effects are important
For 𝛽 = 1 Mott scattering formula 𝑑𝜎
𝑑𝛺 Mott≃
𝑑𝜎
𝑑𝛺 Ruthcos2
𝛩
2
𝑒−
𝑒−
Rutherford with relativistic effects (assuming no nuclear spatial extension):
valid for small Z𝑑𝜎
𝑑𝛺≃
𝑑𝜎
𝑑𝛺Ruth
1 − 𝛽2 sin2𝛩
2
2727
Hofstadter, R., et al., Phys. Rev. 92, 978 (1953).
Nuclear systems cannot be treated as point-like
particles and have finite extension
𝑒−
𝑒−
𝑑𝜎
𝑑𝛺 Mott≃
𝑑𝜎
𝑑𝛺 Ruthcos2
𝛩
2
𝑒−
Or
?
28
Quantum collisions
What ? Why for nuclei ?
2929
We assume that the “entities” interact through a central
potential 𝑉 Ԧ𝑟𝐴 − Ԧ𝑟𝑎
Classical picture
a
A
Incoming beam
Quantum picture
Incoming
Outgoing
Incoming beam
𝐸𝐵 =𝑃𝑎2
2𝑀𝑎𝐸𝐵 =
ℏ2𝑘2
2𝑀𝑎
Trajectories are deterministic
ൿห𝜙𝑖
𝑟՜∞𝑒𝑖𝑘. Ԧ𝑟plane wave
Outgoing spherical wave
ቚ𝜙𝑓𝑟՜∞
𝑓 Θ,𝜑𝑒𝑖𝑘𝑟
𝑟
3030
Few implications of quantum mechanics
➢ Interfering trajectories can be
constructive or destructive.
➢ Impact parameter is not defined in
the quantum world
Trajectories are indistinguishable
𝑏′
𝑏
Impossible in
classical world
Particle-Target motion must be
described by wave functions➢ Quantum interference phenomenon
3131
𝑑𝜎 𝜃,𝜑
𝑑Ω=𝑗𝑓 𝜃, 𝜑
𝑗𝑖
We start from and
ൿห𝜓+ = ൿห𝜙𝑖 + ቚ𝜙𝑓
𝜓𝑘+ Ԧ𝑟
𝑟՜∞𝐴 𝑒𝑖𝑘. Ԧ𝑟 + 𝑓 Θ,𝜑
𝑒𝑖𝑘𝑟
𝑟
Quantum current is defined as
Ԧ𝐽 =ℏ
2𝜇𝑖𝜓𝑘+ ∗∇𝜓𝑘
+ − 𝜓𝑘+∇ 𝜓𝑘
+ ∗
Ԧ𝐽 = Ԧ𝑗𝑖 + Ԧ𝑗𝑓 + Ԧ𝑗int
𝑗𝑓 𝜃, 𝜑 = Ԧ𝐽 ∙ Ƹ𝑟 𝑟−2
𝑗𝑖 = Ԧ𝐽 ∙ Ƹ𝑧
𝑗𝑓 𝜃, 𝜑 = 𝑣 𝐴 2 𝑓 Θ,𝜑 2
𝑗𝑖 = 𝑣 𝐴 2
𝑑𝜎 𝜃, 𝜑
𝑑Ω= 𝑓 Θ,𝜑 2
𝑗int = 0 for θ ≠ 0
gives the optical theorem
We find the cross section to be:Τ1 𝑟 leading
contributions
3232
Ex: scattering of 12C + 12C @ 5 MeV
Since nucleons are fermions One cannot distinguish
the two cases i.e. wf is
antisymetric
This leads to
Rutherford
Exp
Illustration from G. R. Satchler“Introduction to direct reactions”
𝑑𝜎 𝜃,𝜑
𝑑Ω= 𝑓 Θ, 𝜑 + 𝑓 𝜋 − Θ,𝜑 2
causes interferences !
Note that the same happens with Coulomb