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Introduction to Nuclear physics;
The nucleus a complex system
Héloïse Goutte
CEA, DAM, DIF
[email protected]
Héloïse Goutte
CERN Summer student program
2009
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The nucleus : a complex system
I) Some features about the nucleus
discovery
radius
binding energy
nucleon-nucleon interaction
life time
applications
II) Modeling of the nucleus
liquid drop
shell model
mean field
III) Examples of recent studies
exotic nuclei
isomers
shape coexistence
super heavy
IV) Toward a microscopic description of the fission process
Héloïse Goutte
CERN Summer student program
2009
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Some features about the nucleus:
summary
A nucleus is made of Z protons and N neutrons (the nucleons).
A nucleus is characterized by its mass number A = N + Z
and its atomic number Z.
It is written AX.
A nucleus is almost 100000 times
smaller than an atom
ATOM
(10-9 m)
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NUCLEUS
(10-14 m)
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NUCLEON
(10-15 m)
2009
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R(fm)
Nuclear radius
R = 1.25 x A1/3 (fm)
A1/3
The radius increases with A1/3
 The volume increases with the number of particles
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CERN Summer student program
2009
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The nucleon-nucleon interaction
V (MeV)
Proton and neutron interact through
the strong interaction.
r(fm)
The strong interaction is
very intense
of short range
The nuclear interaction is stabilizing the nucleus
Proton –neutron interaction : Vpn > Vnn et Vpp Vnn
PLUS
Coulomb interaction between protons (repulsive)
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CERN Summer student program
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Binding energy
Mass of a given nucleus : M(A,Z) = N Mn + Z Mp – B(A,Z)
B(A,Z) : binding energy
Stable bound system for B > 0 (its mass is lower than the mass of its
components)
Unstable systems : they transform into more stable nuclei
Exponential decay
dN
 N (t )
dt
Half –life T defined as the time for which
the number of remaining nuclei
is half of its the initial value.
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CERN Summer student program
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Different types of radioactivity
A
Z 1
A 1
Z
XN1
X N 1
a
A 4
Z2
bn X
p
A
Z
A 1
N
XN
Z 1
b+,e
A
Z 1
X N 1
X N 2
Neutrons
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How do we experimentally study a nucleus ?
(e- ,e+, p,n, heavy
ions, …)
I ) Elastic and inelastic scattering
II ) Gamma spectroscopy
The level scheme:
the barcode of a nucleus
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II) Modeling of the nucleus
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The nucleus : a liquid drop ?
The nucleus and its features, radii, and binding energies have many
similarities with a liquid drop :
 The volume of a drop is proportional to its number of molecules.
 There are no long range correlations between molecules in a drop.
-> Each molecule is only sensitive to the neighboring molecules.
-> Description of the nucleus in term of a model of a charged liquid drop
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CERN Summer student program
2009
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The liquid drop model
* Model developed by Von Weizsacker and N. Bohr (1937)
It has been first developed to describe the nuclear fission.
* The nucleus is represented by a charged liquid drop.
* The model has been used to predict the main properties of
the nuclei such as:
* nuclear radii,
* nuclear masses and binding energies,
* decay out,
* fission.
* The binding energy of the nuclei is described by
the Bethe Weizsacker formula
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Binding energy for a liquid drop
Mass formula of Bethe and Weizsäcker
2
3
B(A, Z)  a v A  a s A  a c
Z2
A
1
3
(N  Z)2
 aa
A
(for e-e nuclei)

0 

(for o-o nuclei)
av = volume term 15.56 MeV
as = surface term 17.23 MeV
ac = coulomb term 0.697 MeV
aa= asymmetry 23.235 MeV
 = pairing term
12 A
Parameters adjusted to
experimental results
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CERN Summer student program
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Problems with the liquid drop model
1) Nuclear radii
Evolution of mean square radii with
respect to 198Hg as a function of
neutron number.
Light isotopes are unstable nuclei
produced at CERN by use of
the ISOLDE apparatus.
-> some nuclei away from the A2/3 law
Fig. from
http://ipnweb.in2p3.fr/recherche
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CERN Summer student program
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Halo nuclei
I. Tanihata et al., PRL 55 (1985) 2676
I. Tanihata and R. Kanungo, CR Physique (2003) 437
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2) Nuclear masses
E (MeV)
Difference in MeV between experimental masses and masses calculated with
the liquid drop formula as a function of the neutron number
Neutron number
Existence of magic numbers : 8, 20 , 28, 50, 82, 126
Fig. from L. Valentin, Physique subatomique, Hermann 1982
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Two neutron separation energy S2n
S2n : energy needed to
remove 2 neutrons
to a given nucleus (N,Z)
S2n=B(N,Z)-B(N-2,Z)
For most nuclei, the 2n separation energies are smooth functions of
particle numbers apart from discontinuities for magic nuclei
Magic nuclei have increased particle stability and require a larger
energy to extract particles.
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CERN Summer student program
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3) Fission fragment distributions
A heavy and a light fragments
= asymmetric fission
Proton number
Liquid drop : only symmetric fission
Neutron number
Experimental Results :
Two identical fragments
= symmetric fission
K-H Schmidt et al., Nucl. Phys. A665 (2000) 221
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CERN Summer student program
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The nucleus is not a liquid drop : structure effects
There are many « structure effects » in nuclei, that can not be
reproduced by macroscopic approaches like the liquid drop model
There are «magic numbers»
2, 8, 20, 28, 50, 82, 126
and so «magic»
and «doubly magic» nuclei
90
140
Zr
40
50 58 Ce82......
40
20
Ca2 0
208
82
Pb1 2 6 ...100
5 0 Sn1 0 0 ...
-> need for microscopic approaches, for which the fundamental
ingredients are the nucleons and the interaction between them
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CERN Summer student program
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Microscopic description of the atomic nucleus
Nucleus = N nucleons in strong interaction
The many-body problem
(the behavior of each nucleon
influences the others)
Nucleon-Nucleon force
unknown
No complete derivation from the QCD
Can be solved exactly for N < 4
For N >> 10 : approximations
Shell model
• only a small number of
particles are active
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Different forces used depending
on the method chosen to solve the
Many-body problem
Approaches based
on the mean field
• no inert core
• but not all the correlations
between particles are taken
into account
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Quantum mechanics
Nucleons are quantum objects :
Only some values of the energy are available : a discrete number of states
Nucleons are fermions :
Two nucleons can not occupy the same quantum state : the Pauli principle
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 
 
 
 
 
Neutrons
Protons
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The Goeppert Mayer shell model
* Model developped by M. Goeppert Mayer in 1948 :
The shell model of the nucleus describes the nucleons in the nucleus
in the same way as electrons in the atom.
* “In analogy with atomic structure one may postulate that in the nucleus
The nucleons move fairly independently in individual orbits in an average
potential which we assume to have a spherical symmetry” ,
M. Goeppert Mayer, Nobel Conference 1963.
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CERN Summer student program
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Nuclear potential
Nuclear potential deduced from exp :
Wood Saxon potential
or
square well
or
harmonic oscillator
Thanks to E. Gallichet
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CERN Summer student program
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Quantum numbers
Quantum numbers characterizing the
Nucleon states:
The principal quantum number N
The radial quantum number n
The azimutal quantum number l
The spin s (s = ± ½)
N= 2(n-1) +l
Introduction of the angular momentum
j= l ± 1/2
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CERN Summer student program
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Single particle levels in the shell model
-3d5/2
-2g9/2
6 hw
5 hw
3p
2f
-1i13/2
-3p1/2
-3p3/2
-2f5/2
-2f7/2
-1h9/2
126
-1h11/2
+3s1/2
+1d3/2
82
1h
4 hw
3s
2d
+2d5/2
+1g7/2
1g
3 hw
+1g9/2
-1p1/2
-1f5/2
-2p3/2
2p
1f
-1f7/2
28
+1d3/2
+2s1/2
+1d5/2
2 hw
2s
1d
1 hw
1p
-1p1/2
-1p3/2
0 hw
1s
+1s1/2
Isotropic
Square well (n,l)
50
20
8 Magic
2 Numbers
Isotropic square well
+ spin-orbit (n,l,j)
j = l +/- 1/2
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CERN Summer student program
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Beyond this “independent particle shell model”
Satisfying results for magic nuclei :
ground state and low lying excited states
Problems :
• Neglect of collective deformation, vibration, rotation
• Same potential for all the nucleons and for all the configurations
• Independent particles
•Improved shell model (currently used):
The particles are not independent : due to their interactions with
the other particles they do not occupy a given orbital but a sum of
configurations having a different probability.
-> definition of a valence space where the particles are active
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CERN Summer student program
2009
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The shell model space
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CERN Summer student program
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The self consistent mean field approach
Main assumption: each particle is interacting with an average field
generated by all the other particles : the mean field
The mean field is built from the individual excitations between the
nucleons
Self consistent mean field : the mean field is not fixed. It depends on
the configuration.
No inert core
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CERN Summer student program
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The self consistent mean field approach
The Hartree Fock method
The basis ingredient is the Hamiltonian which governs the dynamics of the
individual nucleons (equivalent to the total energy in classical physics)

pi2 1 A eff
H 
  vij
2
M
2 i  j 1
i 1
A
Effective force
( x1, x2 ,..., xA )
Wave function
= antisymmetrized product of A

orbitals of the nucleons i ( xi ) with xi  (ri , i , i )
Orbitals are obtained by minimizing the total energy of the nucleus
E
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 H

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The phenomenological
effective finite-range Gogny force
P : isospin exchange operator
P : spin exchange operator
 r1 - r2 2 
 Wj  B jP - H jP - M jP P 
v12   exppj 

j 1

Finite range central term
  a  
 t 3 1  x0 Ps  r1  r2  r1  r2  Density dependent term

  


 iWls 12. r1  r2  12 . 1   2 
Spin orbit term
2
e2
 1  2t1 z 1  2t 2 z   
r1  r2
Coulomb term
back
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CERN Summer student program
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The Hartree Fock equations
Hartree-Fock equations
  h2 2

  U HF (a )i ( xi )  e ii ( xi )

2
M


(A set of coupled Schrodinger equations)
Single particle wave functions
Hartree-Fock potential
Self consistent mean field :
the Hartree Fock potential depends on the solutions
(the single particle wave functions)
-> Resolution by iteration
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Resolution of the Hartree Fock equations
Trial single particle wave function
i ( xi )
Effective interaction
Calculation of the HF
potential
U HF (a )
  h2 2

Resolution of the HF equations 
  U HF (a )i ( xi )  e ii ( xi )
 2M

New wave functions
i ( xi )
Test of the convergence
Calculations of the properties of the nucleus in its ground state
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CERN Summer student program
2009
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Deformation
We can “measure” nuclear deformations as the mean values of the
mutipole operators Q̂
q 
 Qˆ  

Spherical Harmonic
If we consider the isoscalar axial quadrupole operator Qˆ 20  r 2 Y20
We find that:
http://www-phynu.cea.fr
Most ot the nuclei are deformed in their ground
state
Magic nuclei are spherical
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CERN Summer student program
2009
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Results
g.s deformation predicted
with HFB using the Gogny
force
http://www-phynu.cea.fr
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CERN Summer student program
2009
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Constraints Hartree-Fock-Bogoliubov
calculations
We can impose collective deformations and test the response of the
nuclei
  qi  Hˆ   iQˆi  N Nˆ  Z Zˆ qi   0
i
with
qi  Nˆ ( Zˆ ) qi   N (Z)
qi  Qˆ i qi   q i
Where ’s are Lagrange parameters
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2009
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Potential energy curves
Example : fission barrier
Energy (MeV)
First and second barriers
Isomeric well
Ground state
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Quadrupole Deformation
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What are the most commonly used constraints ?
What are the problems
with this deformation ?
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Potential energy landscapes
Deformations pertinent for fission:
Elongation
Asymmetry
…
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Evolution of s.p. states with deformation
New gaps
Energy (MeV)
126
108
116
98
82
96
Deformation
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From microscopic calculations
154Sm
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Beyond the mean field
Introduction of more correlations : two types of approaches
Random Phase Approximation (RPA)
Generator coordinate Method (GCM)
Coupling between HFB ground state
and particle hole excitations
Introduction of large amplitude
correlations
Give access to
a correlated ground state and to the excited states
Individual excitations and collective states
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Results from beyond mean field calculations
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The nuclear shape : a pertinent information ?
Spherical nuclei
«vibrational» spectrum
6+
4+
2+
E  J
0+
Deformed nuclei
«rotational» spectrum
6+
4+
2+
E  J(J +1)
0+
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Two examples of spectra
160Gd
148Sm
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Change of the shape of the nuclei
along an isotopic chain
Sm
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Angular velocity of a rotating nucleus
For a rotating nucleus, the energy of a level is given by* :
With J the moment of inertia
We also have
so
With
To compare with a wash machine: 1300 tpm
* Mécanique quantique by C. Cohen-Tannoudji, B. Diu, F. Laloe)
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CERN Summer student program
2009
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Modeling of the nuclei :
Summary
• Macroscopic description of a nucleus : the liquid drop model
• Microscopic description needed:
the basic ingredients are the nucleons and the interaction between them.
• Different microscopic approaches : the shell model and the mean field
• Many nuclei are found deformed in their ground states
• The spectroscopy strongly depends on the deformation
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CERN Summer student program
2009
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