1. Introduction to Nuclear physics; The nucleus a complex system
Héloïse Goutte
GANIL
Caen, France 1

2. 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 2

3. Nuclear radius R = 1.25 x A1/3 (fm) The radius increases with A1/3
R(fm)
R = 1.25 x A1/3 (fm)
A1/3
The radius increases with A1/3
 The volume increases with the number of particles 4

4. 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
Half –life T defined as the time for which
the number of remaining nuclei
is half of its the initial value. 6

5. II) Modeling of the nucleus10

6. 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 11 11

7. 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 12

8. Binding energy for a liquid drop
(for e-e nuclei)
Mass formula of Bethe and Weizsäcker
(for o-o nuclei)
av = volume term MeV
as = surface term MeV
ac = coulomb term MeV
aa= asymmetry MeV
 = pairing term
Parameters adjusted to
experimental results 13

9. 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 14

10. 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
2) Nuclear masses
Difference in MeV between experimental masses and masses calculated with
the liquid drop formula as a function of the neutron number
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. 3) 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

14. 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
-> 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

15. 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 forces used depending
on the method chosen to solve the
Many-body problem 20

16. 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

17. 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 …” ,
M. Goeppert Mayer, Nobel Conference 1963. 22

18. Nuclear potential deduced from exp :
Wood Saxon potential or
square well
harmonic oscillator
Thanks to E. Gallichet 23

19. 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 24

20. Single particle levels in the shell model
6 hw
-1i13/2
126 3p
-3p1/2
-3p3/2 2f
-2f5/2
5 hw
-2f7/2
-1h9/2 1h
-1h11/2 82 3s
+3s1/2
+1d3/2 2d
4 hw
+1g7/2
+2d5/2 1g 50
+1g9/2
-1p1/2 2p
-1f5/2
3 hw
-2p3/2 28 1f
-1f7/2
+1d3/2 20 2s
+2s1/2
2 hw 1d
+1d5/2 8
0 hw
1 hw 1p
-1p1/2
Magic
Numbers
-1p3/2 2 1s
+1s1/2
Square well (n,l)
Square well
+ spin-orbit (n,l,j)
j = l +/- 1/2 25

21. How do you describe the ground state ?
126
-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
-1h11/2 82
+3s1/2
+1d3/2
+1g7/2
+2d5/2 50
+1g9/2
-1p1/2
-1f5/2
-2p3/2 28
-1f7/2
+1d3/2 20
+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

22. How do you 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

23. 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

24. The shell model space 27

25. 28

26. 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 29

27. 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)
Effective force
Wave function = antisymmetrized product of A
orbitals of the nucleons with
Orbitals are obtained by minimizing the total energy of the nucleus 30

28. effective finite-range Gogny force
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
back 31

29. The Hartree Fock equations
(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 32

30. Resolution of the Hartree 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
Calculations of the properties of the nucleus in its ground state 33

31. 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 ot the nuclei are deformed in their ground state
Magic nuclei are spherical 34

32. Results g.s deformation predicted with HFB using the Gogny force 35

33. Constraints Hartree-Fock-Bogoliubov
calculations
We can impose collective deformations and test the response of the nuclei
with
Where ’s are Lagrange parameters 36

34. Example : fission barrier
Potential energy curves
Example : fission barrier
First and second barriers
Energy (MeV)
Isomeric well
Ground state
Quadrupole Deformation 37

35. What are the most commonly used constraints ?
What are the problems
with this deformation ? 38

36. Deformations pertinent for fission:
Potential energy landscapes
Deformations pertinent for fission:
Elongation
Asymmetry … 39

37. Evolution of s.p. states with deformation
New gaps
126
108
116
Energy (MeV) 98 82 96
Deformation 40

38. From microscopic calculations
154Sm 41

39. Beyond the 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

40. Results from beyond mean field calculations43

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

42. Two examples of spectra
160Gd
148Sm 45

43. Change of the shape of the nuclei along an isotopic chainSm 46

44. 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

45. 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 48