1. Introduction to Nuclear Physics
2/3
S.PÉRU

2. The nucleus : a complex system
I) Some features about the nucleus
discovery
radius, shape
binding energy
nucleon-nucleon interaction
stability and life time
nuclear reactions
applications
II) Modeling of the nucleus
liquid drop
shell model
mean field
III) Examples of recent studies
figures of merit of mean field approaches
exotic nuclei
isomers
shape coexistence
IV) Toward a microscopic description of the fission process

3. Modeling of the nucleus
nucleus = A nucleons in interaction
2 challenges
Nuclear Interaction inside the nuclei (unknown)
N body formalism
The liquid drop model : global view of the nucleus associated to a quantum liquid.
The Shell Model : each nucleon is independent in a attractive potential.
« Microscopic » methods ~ Hartree-Fock , BCS ,Hartree-Fock-Bogoliubov :
The nuclear structure is described within the assumption that each nucleon is interacting with an average field generated by all the other nucleons.

4. The nucleus and its features, radii, and binding energies have many
The nucleus is a charged quantum liquid.
Quantum : The wave length of the nucleons is large enough with respect to the size of the nucleons to vanish trajectory and position meaning.
Liquid : Inside the nucleus nucleons are like water molecules.
They roll “ones over ones” without going outside the “container”.
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

5. The liquid drop Model developed by Von Weizsacker and N. Bohr (1937)
It has been first developed to describe the nuclear fission.
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.
Equilibrium shape
spherical
Density ρ0
Volume ≈ A
R= r0 A 1/3 r r R R
The binding energy of the nuclei is described by the Bethe-Weiszaker formula

6. Bethe and Weizsacker formula
Paring terms
+ even-even
- odd-odd
0 odd-even and even-odd
Binding term :
volume av
Unbinding terms :
Surface as ,
Coulomb ac ,
Asymmetry aa

7. Binding energy per nucleon

8. Problems with the liquid drop model
1) Fission fragment distributions
A heavy and
a light fragments
= asymmetric fission
Liquid drop : only symmetric fission
Proton number
Neutron number
Experimental Results :
K-H Schmidt et al., Nucl. Phys. A665 (2000) 221
Two identical fragments
= symmetric fission 18

9. Problems with the liquid drop model
2) 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 14

10. Problems with the liquid drop model
Halo nuclei
I. Tanihata et al., PRL 55 (1985) 2676
I. Tanihata and R. Kanungo, CR Physique (2003) 437 15

11. Fig. from L. Valentin, Physique subatomique, Hermann 1982
3) Nuclear masses
Difference in MeV between experimental masses and masses calculated with
the liquid drop formula as a function of the neutron number
Nuclear
shell effects
E (MeV)
Neutron number
Existence of magic numbers : 8, 20 , 28, 50, 82, 126
Fig. from L. Valentin, Physique subatomique, Hermann 1982 16

12. 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. 17

13. The nucleus is not a liquid drop :
Shell effects
There are many « structure effects » in nuclei, that can not be
reproduced by macroscopic approaches like the liquid drop model.
-> need for microscopic approaches, for which the fundamental
ingredients are the nucleons and the interaction between them
There are «magic numbers» 2, 8, 20, 28, 50, 82, 126
and so «magic»
and «doubly magic» nuclei 19

14. Nucleus = N nucleons in strong interaction
Microscopic description of the atomic nucleus
Nucleus = N nucleons in strong interaction
The many-body problem
(the behavior of each nucleon
influences the others)
Can be solved exactly for N < 4
For N >> 10 : approximations
Shell Model
only a small number of
particles are active
Approaches based on the Mean Field
no inert core
but not all the correlations between particles are taken
into account
Nucleon-Nucleon force
unknown
Different effective forces used
Depending on the method chosen
to solve the many-body problem 20

15. 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 
Neutrons
Protons 21

16. Shell Model Model developed 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 …” ,
M. Goeppert Mayer, Nobel Conference 1963.

17. Shell Model U (r ) Schrödinger equation
Energy (MeV)
U (r ) R
r (fm)
Schrödinger equation
Wave function φ and energy ε for each nucleon
Wave function ψ and nuclear binding energy E R
Features of the nucleus in his ground state and his excited ones r

18. Shell Model : potential
Nuclear potential deduced from exp :
Wood Saxon potential or
square well
harmonic oscillator

19. Shell Model : potential
spin orbit effect

20. Shell Model : how describe the ground state ?
126
-1i13/2
-3p1/2
-3p3/2
-2f5/2
+1s1/2
-1p3/2
-1p1/2 2 8
Ex: Z=10
+1d5/2
+2s1/2
+1d3/2 20
Ex: N=20
-2f7/2
-1h9/2 82
-1h11/2
+3s1/2
+1d3/2
+1g7/2
+2d5/2 50
+1g9/2
-1p1/2
-1f5/2
-2p3/2 28
-1f7/2 20
+1d3/2
+2s1/2
+1d5/2 8
-1p1/2
-1p3/2 2
+1s1/2
For a nucleus with A nucleons you fill the A lowest energy levels, and
the energy is the sum of the energy of the individual levels

21. Shell Model : how describe excited states ?
-1p3/2
-1p1/2 2 8
Ex: Z=10
+1d5/2
+2s1/2
+1d3/2 20
Ex: N=20
-1f7/2
Ground state
+1s1/2
-1p3/2
-1p1/2 2 8
Ex: Z=10
+1d5/2
+2s1/2
+1d3/2 20
Ex: N=20
-1f7/2
Excited state :
you make a particle-hole excitation.
You promote one particle to a higher
energy level

22. Beyond this “independent particle Shell Model
Satisfying results for magic nuclei :
ground state and low lying excited states
Problems :
Neglect of collective excitations
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 26

23. Beyond this “independent particle Shell Model
proton
neutron 26

24. many nucleons  effective interaction
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.
Nuclear interaction
2 nucleons bare force
many nucleons  effective interaction
Neptune FG
Feffective
Soleil
Flibre
Uranus
Self consistent mean field : the mean field is not fixed. It depends on the configuration.
No inert core

25. Jacques Dechargé
Jacques Dechargé

26. Finite range for pairing treatment
The phenomenological effective finite-range Gogny force
P : isospin exchange operator
P : spin exchange operator
Finite range
central term
Density dependent term
Spin orbit term
Coulomb term
Finite range for pairing treatment

27. Mean field approach : Hatree-Fock method
Hartree-Fock equations
(A set of coupled Schrodinger equations)
For more formalism see
“The nuclear many body problem”,
P. Ring and P. Schuck
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

28. Resolution of the Hatree-Fock equations
Trial single particle wave function
Effective interaction
Calculation of the HF potential
Resolution of the HF equations
New wave functions
Test of the convergence
Jacques Dechargé
Calculations of the properties of the nucleus in its ground state

29. Hatree-Fock method : deformation
We can “measure” nuclear deformations as the mean values of the
mutipole operators
Spherical Harmonic
If we consider the isoscalar axial quadrupole operator
We find that:
Most of the nuclei are deformed in their ground state
Magic nuclei are spherical 34

30. Constraints Hatree-Fock-bogoliubov calculations
We can impose collective deformations and test the response of the nuclei:
with
Where ’s are Lagrange parameters. 36

31. Constraints Hatree-Fock-bogoliubov calculations : results
g.s deformation predicted
with HFB using the Gogny
force 35

33. Constraints Hatree-Fock-bogoliubov calculations : What are the most commonly used constraints ?
What are the problems
with this deformation ? 38

34. Deformations pertinent for fission:
Constraints Hatree-Fock-bogoliubov calculations : Potential energy landscapes
Deformations pertinent for fission:
Elongation
Asymmetry … 39

35. Evolution of s.p. states with deformation
154Sm
New gaps

36. Hatree-Fock-bogoliubov calculations with blocking
Particle-hole excitations  one (or two, three, ..) quasi-particles curves

37. Beyond mean field …
Introduction of more correlations : two types of approaches
Random Phase Approximation (RPA)
Coupling between HFB ground state
and particle hole excitations
Generator coordinate Method (GCM)
Introduction of large amplitude
correlations
Give access to
a correlated ground state and to the excited states
Individual excitations and collective states 42

38. Beyond mean field … with GCM
(GCM+GOA 2 vibr. + 3 rot.)
= 5 Dimension Collective Hamiltonian
5DCH

39. Beyond mean field … with RPA
Monopole
Dipole
S. Péru, J.F. Berger, and P.F. Bortignon, Eur. Phys. Jour. A 26, (2005)

40. The nuclear shape : spectrum ?
«vibrational» spectrum
Spherical nuclei 6+ 4+ 2+ 0+
Deformed nuclei
«rotational» spectrum 0+ 4+ 6+ 2+ 44

41. 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 47

42. Modeling 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, the mean field and beyond
Many nuclei are found deformed in their ground states
The spectroscopy strongly depends on the deformation 48